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August 11, 2016

A Survey of Magnitude

Posted by Tom Leinster

The notion of the magnitude of a metric space was born on this blog.
It’s a real-valued invariant of metric spaces,
and it came about as a special case of a general definition of the
magnitude of an enriched category (using Lawvere’s amazing
observation that metric spaces are usefully viewed as a certain kind of enriched category).

Anyone who’s been reading this blog for a while has witnessed the
growing-up of magnitude, with all the attendant questions, confusions,
misconceptions and mess. (There’s an incomplete list of past posts
here.)
Parents of grown-up children are apt to forget that their
offspring are no longer helpless kids, when in fact they have a mortgage and children of their own. In the same way, it would be
easy for long-time readers to have the impression that the theory of
magnitude is still at the stage of resolving the basic questions.

Certainly there’s still a great deal we don’t know. But by now there’s also lots
we do know, so Mark Meckes and I recently wrote a survey
paper:

Here I’ll tell you some of the highlights: ten things we
used not to know, but do now.

Before I give you the highlights, I can’t resist repeating the definition
of magnitude, as it’s so very simple.

Let AA be a finite metric space. Denote by ZAZ_A the square matrix whose
rows and columns are indexed by the points of AA, and with entries ZA(a,b)=e−d(a,b)Z_A(a,
b) = e^{-d(a, b)}. Assuming that ZAZ_A is invertible, the magnitude
of AA is

|A|=∑a,b∈AZA−1(a,b)
\left|A\right| = \sum_{a, b \in A} Z_A^{-1}(a, b)

— the sum of all the entries of the inverse matrix ZA−1Z_A^{-1}.

This definition immediately prompts the questions: why should ZAZ_A be
invertible? And how do you define magnitude for metric spaces that aren’t
finite? The answers appear right at the beginning of the list below.

So, here are ten things we now know about magnitude that once upon a time we didn’t. They’re all covered in the survey paper, where you
can also find references.

Magnitude is well-defined (that is, ZAZ_A is invertible) for every finite
subset of Euclidean space. In fact, ZAZ_A is not only invertible but
positive definite when A⊆ℝnA \subseteq \mathbb{R}^n.

There is a canonical way to extend the definition of magnitude from
finite spaces to compact spaces, as long as they’re “positive definite” (meaning that
the matrix ZAZ_A is positive definite for every finite subset AA).
For instance, this includes all compact subspaces of ℝn\mathbb{R}^n.

What I mean by “canonical” is that there are several different ways
that you might think of extending the definition, and they all give the
same answer. For instance, you might define the magnitude of a compact
positive definite space AA as the supremum of the magnitudes of its finite
subsets. Or, you might take some sequence (Bn)(B_n) of finite subsets of AA
whose union is dense in AA, “define” |A|=limn→∞|Bn|\left|A\right| = \lim_{n \to \infty}
\left|B_n\right|, and hope that none of the many things that could go wrong
with this “definition” do go wrong. Or, you might abandon finite
approximations altogether and try to take a more direct, analysis-based
approach. Mark showed that these approaches all work and all give the same
number.

When you introduce a scale factor tt, the magnitude |tA|\left| t A \right|
of a compact set A⊆ℝnA \subseteq \mathbb{R}^n is a continuous function of tt,
taking values in [1,∞)[1, \infty) (assuming A≠∅A \neq \emptyset). This is
called the magnitude function of AA.

The magnitude function of any finite metric space knows its cardinality.
In fact, it is increasing for sufficiently large tt and converges to the cardinality as t→∞t \to \infty. This
illustrates the idea that the magnitude of a finite space is the “effective
number of points”.

The magnitude function of a compact subset of ℝn\mathbb{R}^n knows its volume.
Specifically, Juan-Antonio Barceló and Tony Carbery
showed that for compact A⊆ℝnA \subseteq
\mathbb{R}^n,

The magnitude function of a compact subset of ℝn\mathbb{R}^n also knows
its Minkowski dimension. (Minkowski dimension is one of the more important
notions of fractional dimension; it’s typically equal to the Hausdorff
dimension.) Specifically, Mark showed that the Minkowski dimension of a
compact set A⊆ℝnA \subseteq \mathbb{R}^n is equal to the growth of the
magnitude function, meaning that there are constants c,C>0c, C \gt 0 such that

c<|tA|tdimA<C
c \lt \frac{\left|t A\right|}{t^{\dim A}} \lt C

for all t≫0t \gg 0.

There’s an exact formula for the magnitude of the sphere of any dimension, with the
geodesic metric. This was found by Simon Willerton.

There’s also an exact formula for the magnitude of any odd-dimensional
Euclidean ball, computed by Barceló and Carbery. This formula raises lots
of interesting questions, and I may say more about it in a future post.

The magnitude function of a convex body in ℝn\mathbb{R}^n with the
taxicab metric (i.e. the metric induced by the 1-norm) is a polynomial.
Its degree is the dimension of the body, and the coefficients are certain
geometric measures of the body — e.g. up to a known factor, the top
coefficient is the volume.

There are various asymptotic formulas for the magnitude functions of
other spaces. For instance, Simon showed that the magnitude function of a
homogeneous Riemannian nn-manifold is asymptotically a polynomial of
degree nn whose top coefficient is proportional to the volume and whose
(n−2)(n - 2)th coefficient is proportional to the total scalar curvature. And
Simon and I found various instances in which an asymptotic
inclusion-exclusion principle holds: e.g. for the ternary Cantor set CC,

corresponding to the fact that 3tC3 t C is the disjoint union of 2
copies of tCt C.

You can find more highlights, plus all the details I’ve left out, in our
survey.

There are also a couple of
important developments that we don’t cover:

The magnitude of a graph, and the corresponding homology theory. You can
view a graph as a metric space: the points are the vertices, and distances
are shortest path-lengths, giving all edges length 11. The study
ofmagnitudeforgraphs
has a special flavour, largely because distances in
graphs are always integers. But most excitingly, it is just the shadow of a
graded homology theory for graphs, developed
by Richard Hepworth and Simon Willerton. Magnitude is the Euler characteristic of
that homology theory, in the same way that the Jones polynomial is the Euler
characteristic of Khovanov homology for links. For instance, the product
formula for magnitude is a corollary of a Künneth theorem in homology
— a decategorification of it, if you like. Similarly,
a certain inclusion-exclusion formula for magnitude is the decategorification of a
Mayer-Vietoris theorem for magnitude homology.

There is an intimate connection between magnitude and maximum entropy
— or more exactly, the exponential of entropy, which I like to call
diversity. Mark and I wrote this up
separately a little while ago. In
fact, Mark used the relationship between magnitude and maximum diversity to
prove the result on Minkowski dimension that I mentioned earlier.

Incidentally, I know of just one other categorically-minded paper with
the words “geometric measure theory” in the title, and it’s one of my
favourite papers of all time:

Re: A Survey of Magnitude

Thanks! As I hope is clear, a lot of this cool stuff was done not by me but by Mark Meckes and Simon Willerton, with a decisive recent contribution from Juan-Antonio Barceló and Tony Carbery.

For a metric space AA with up to four points, ZAZ_A is always invertible. But there’s a five-point space that isn’t. Take the complete bipartite graph K3,2K_{3,2}:

This gives rise to the five-point metric space AA whose points are the vertices of this graph and whose distances are shortest path-lengths (and edges have length 1).
Now scale AA by a factor of u=log2u = \log\sqrt{2}. Then ZuAZ_{u A} is not invertible. In fact, here’s the magnitude function of AA:

You can see it has a singularity, which is at t=log2t = \log\sqrt{2}.

Although the magnitude function of a finite metric space can have some singularities, it’s not too hard to prove that there are only finitely many of them. (That’s part of Proposition 2.2.6 here.) Apart from those, it’s analytic.

Re: A Survey of Magnitude

Hi Jamie,

I don’t really believe that anything in mathematics is pathological. There’s just stuff we understand and stuff we don’t.

It can seem surprising at first that one particular scale can play a special role for a metric space. But it’s actually quite common. We’ve been talking about persistent homology a bit recently, so here’s an example of that flavour.

Take the metric space XX consisting of ten points spaced equally around a circle of radius rr in ℝ2\mathbb{R}^2. Around each point, draw the disk of radius ε\varepsilon, where ε\varepsilon is small (at least to begin with). Consider what happens to the homotopy type of the union of the disks as ε\varepsilon grows.

At first, the union of the ten disks is disjoint, so it has the homotopy type of ten points. But for a certain value ε0\varepsilon_0 of ε\varepsilon, each disk meets its neighbour and their union suddenly acquires the homotopy type of a circle. Then later, when ε\varepsilon reaches rr, the disks all merge into one, and thereafter the homotopy type is that of a point.

So associated with our original finite metric space are two special scales, ε0\varepsilon_0 and rr, where the two “catastrophes” (changes of homotopy type) occur.

I can’t give any such appealing interpretation of the special role of the scale log2\log\sqrt{2} for this particular space. In fact, metric spaces corresponding to graphs are about as ungeometric as it’s possible to be. If I’m ever wondering whether a true statement about subsets of ℝn\mathbb{R}^n is also true for general metric spaces, the first place I’ll look for counterexamples is in the class of graphs. From that point of view, maybe it’s not surprising that there’s no easy geometric explanation here.

What do these singularities ‘mean’?

I don’t know, and I’d really like to. For instance, the graph

has magnitude function

5+5q−4q2(1+q)(1+2q)
\frac{5 + 5q - 4q^2}{(1 + q)(1 + 2q)}

where q=e−tq = e^{-t}. (For graphs, it’s useful to make that substitution so that the magnitude is always a rational function.) It therefore has singularities at q=−1q = -1 and q=−1/2q = -1/2. How should we interpret those numbers? I spent a while thinking about this general question, and I simply don’t know.

Is the magnitude still defined for these metric spaces by some other method?

Their magnitude as a real number probably isn’t defined, but if you’ll allow the magnitude to live elsewhere then you’re probably OK.

Let AA be a finite metric space (which for convenience I’ll assume satisfies the traditional axiom of symmetry). Note that [0,∞)[0, \infty) is a rig in the obvious way, so we can talk about commutative algebras over it. Let RAR_A be the commutative [0,∞)[0, \infty)-algebra generated by one generator waw_a for each a∈Aa \in A, subject to the equation

∑be−d(a,b)wb=1
\sum_b e^{-d(a, b)} w_b = 1

for each a∈Aa \in A. Define [A][A], the “formal magnitude” of AA, to be the element ∑awa\sum_a w_a of RAR_A.

If there exists a [0,∞)[0, \infty)-algebra homomorphism ϕ:RA→ℝ\phi \colon R_A \to \mathbb{R} then ϕ([A])∈ℝ\phi([A]) \in \mathbb{R} is independent of the choice of ϕ\phi. (That’s true for any [0,∞)[0, \infty)-algebra, not just ℝ\mathbb{R}.) Moreover, in that case, ϕ([A])\phi([A]) is the ordinary magnitude |A||A| of AA. So the question shifts from “Is the magnitude defined?” to “In what algebras is the magnitude defined?”

Re: A Survey of Magnitude

Are these particular numbers like log2\log \sqrt{2} actually determined in an invariant way, or are they just an artifact of the particular choice of the base ee for the exponentials appearing in the matrix ZAZ_A? I mean, of course ee is a somewhat canonical choice of a base for exponentials, but it’s still a choice, and in some contexts (e.g. the homological one) it seems more natural to take the base to be a formal variable.

Re: A Survey of Magnitude

Well, if you replace ee by some other base bb then log2\log\sqrt{2} (meaning loge2\log_e \sqrt{2}) gets replaced by logb2\log_b\sqrt{2} … but I think you know that! In that sense, anyway, log2\log\sqrt{2}is an artifact of the base chosen. But its exponential 2\sqrt{2} isn’t: that’s a number intrinsic to the metric space, unaffected by our choice of base for magnitude.

In the context of graph magnitude, we do indeed replace ee (or rather e−1e^{-1}) by a formal variable, qq. Because distances in graphs are integers, we only ever get integer powers of qq.

We could attempt to make a similarly-flavoured theory of magnitude for arbitrary finite metric spaces, and that would mean manipulating formal expressions involving powers qℓq^\ell where ℓ\ell is a positive real number. Once more, that’s something Aaron Greenspan and I played with for a while without getting too far. Is there an existing theory of such formal expressions, a cousin of the theory of formal power series?

At some stage, it’s interesting to actually realize magnitude as a real number rather than some formal expression, because that’s how you extract real invariants such as volume and Minkowski dimension from magnitude. But one could take the position that this realization process should be left until the end.

Re: A Survey of Magnitude

Re: A Survey of Magnitude

Thinking about Hahn series a bit more, it seems to me that they ought to work pretty well at least up to a point. The entries in ZAZ_A are just real-number powers of qq, which live in a Hahn series field like k[qℝ]k[q^{\mathbb{R}}] where kk is any field. Since Hahn series form a field, we can use the usual formulas to invert a matrix over that field, and as long as the characteristic of kk is 00 the matrix of a metric space should always be invertible over this field. Thus, the magnitude of a finite metric space should always be defined as a Hahn series over ℚ\mathbb{Q}.

It’s less clear whether the formal calculation works over Hahn series to identify the coefficients of the Hahn-series magnitude with the desired Euler characteristics of the magnitude chain complex. But it seems that there’s at least a chance.

Also it’s not entirely clear (at least, not to me) how to “evaluate” a general Hahn series at a value of the variable — or, rather, I can guess how one would try to do it, but I don’t know anything about what sort of convergence properties one can hope for. But perhaps if we do some reading about Hahn series we can find out.

Re: A Survey of Magnitude

Here’s a slightly different take on Tom’s response to Jamie’s question. The definition of magnitude as stated above involves an arbitrary choice of scale. We can avoid this unpleasant arbitrariness in at least three ways, which are subtly different perspectives on the same thing:

Instead of looking just at AA, look at the whole family {tA:t∈(0,∞)}\{t A : t \in (0, \infty)\} of rescalings of AA, and then take the magnitudes of those. This is how Tom defined the magnitude function above, and the last part of Jamie’s question seems to be coming at the issue from this perspective: what’s going on with this one particular rescaling of K3,2K_{3,2}?

Use all possible scales to define magnitude, so that the magnitude of AA is a (partially-defined) function instead of a real number. Forget the definitions of magnitude and magnitude function above, and suppose we instead start by defining a family of matrices ZA,t(a,b)=e−td(a,b)Z_{A, t}(a,b) = e^{-t d(a,b)} for t∈(0,∞)t \in (0,\infty). Now define the magnitude function of AA by
MA(t)=∑a,b∈AZA,t−1(a,b)
M_A(t) = \sum_{a,b \in A} Z_{A, t}^{-1}(a, b)
for all t∈(a,b)t \in (a,b) for which ZA,tZ_{A,t} is invertible. Notice the difference? Of course MA(t)=|tA|M_A(t) = | t A |, so this gives the same thing as before, but in this approach, I never mentioned rescaling the space AA itself.

Replace the base of the exponent with a formal variable, as Mike suggests. As Tom alludes to, whether perspective 2 or 3 is more useful seems to depend on context. For graphs, 3 is nice because then magnitude is a rational function. For infinite spaces, 2 is nice because we can use analytic tools, and it’s not clear whether the magnitude function is at all nice algebraically.

From either of the latter two perspectives, there’s no reason even to ask “What’s so pathological about (log2)K3,2(\log \sqrt{2}) K_{3,2}?” On the other hand, Jamie’s first question:

What do these singularities ‘mean’?

becomes even more interesting. The fact that the magnitude function of K3,2K_{3,2} has a singularity at log2\log \sqrt{2} depends on the fact that we say the edges of K3,2K_{3,2} have length 1; change that basic length and the location of the singularity changes. But the facts that there is a singularity in the magnitude function, and that there is only one in (0,∞)(0,\infty) (but three if you let the base be a formal variable, or look at the meromorphic continuation), are intrinsic facts about K3,2K_{3,2}, and about any rescaling of it. In the geometry of metric spaces, people are usually interested in properties which are invariant (or at least behave predictably) under rescaling, and these singularities give us a lot of such properties. (Going farther, from the formal-variable/rational-function perspective, we could start looking at degrees of poles, residues, etc.)

So what does it “mean” that the magnitude function of K3,2K_{3,2} has a singularity in (0,∞)(0,\infty)? What does it “mean” that, if you broaden your perspective sufficiently, it has three singularities? What does it “mean” that those singularities are they types that they are?

I don’t know answers to all those questions. One thing we know is that if the magnitude function of a finite metric space AA has a singularity in (0,∞)(0,\infty), then AA is not of “negative type”, which implies in particular that AA cannot be isometrically embedded in a Euclidean space, or in L1L^1 (a stronger statement, since L2L^2 embeds isometrically in L1L^1). These facts can be proved without using magnitude, but they’re enough to convince me that Jamie’s question is worth thinking about.

Re: A Survey of Magnitude

From the homological perspective, the magnitude function looks more like a generating function than like a natively analytic object: we have some sequence of numbers, and for some reason we decide to put them as the coefficients in a (perhaps formal) power series. So what does it “mean” in general if the generating function of a combinatorial sequence (converges and) has singularities?

Re: A Survey of Magnitude

Nice! The key phrase seems to be analytic combinatorics: I can’t make much sense out of the wikipedia article, but the first paragraph sounds exactly right:

analytic combinatorics… uses the internal structure of the objects to derive formulas for their generating functions and then complex analysis techniques to get asymptotics.

I found it easier to start with lecture 04 from the course you linked to. For instance, on page 59 we find (rephrased)

If hh is meromorphic in a disc and analytic at 0, and has a unique pole α\alpha closest to the origin, then α∈ℝ\alpha\in\mathbb{R} and the coefficients of the power series expansion h(z)=∑N≥0hNzNh(z) = \sum_{N\ge 0} h_N z^N of hh at 00 grow like cα−NNM−1c \alpha^{-N} N^{M-1} where MM is the order of the pole α\alpha.

So it seems that the singularities of the magnitude function of a graph are telling us something about the growth of the Euler characteristics of its magnitude homology as a function of ℓ\ell.

Re: A Survey of Magnitude

Re: A Survey of Magnitude

One difference between approaches (1) and (2) is that at least in principle, (2) makes sense for any t∈ℂt\in\mathbb{C}, whereas (1) only makes sense for positive tt.

Complex values of tt seem like they might be useful for showing that the magnitude homology determines the entire magnitude function. If fAf_A is the magnitude function of AA, then a priori the magnitude homology of AA determines (via Euler characteristic) a power series expansion of fA(−log(q))f_A(-log(q)) about q=0q=0, which (given that fAf_A is real analytic) determines the magnitude function only on an interval around t=∞t=\infty stretching back to the first singularity. But once we know fAf_A is complex analytic with isolated singularities (since it is a quotient of two holomorphic functions, as noted in the proof of Prop 2.2.6), its value in one disc determines its value everywhere else by analytic continuation.

Re: A Survey of Magnitude

Is there any way to generalize the Hepworth–Willerton homology from graphs to general finite metric spaces, or any way to see that it doesn’t generalize?

Is there any way to use the magnitude for graphs, or this homology theory, to solve any puzzles of the sort graph theorists tend to like — either existing ones, or new ones? One reason people liked the Jones polynomial is that it cracked one of the century-old Tait conjectures, and one reason people like Khovanov homology is that it gave purely combinatorial proof of the Milnor conjecture. Of course the other big reason people like these invariants is that they’re connected to quantum groups and gauge field theory, but there’s a certain kind of knot theorist, or graph theorist, who asks “what will this do for me?” It’s probably not good to let that attitude drive your research unless you have that attitude… but nonetheless it’d be nice to use magnitude-related ideas to solve some puzzles.

Re: A Survey of Magnitude

Is there any way to generalize the Hepworth–Willerton homology from graphs to general finite metric spaces?

I would love it if someone found a way to do this.

Aaron Greenspan and I spent a while trying to do it ourselves, but we didn’t get too far. Then recently, I was at a fantastic applied topology conference where all the talk of persistent homology revived my urge to do it. (As you know, persistent homology is itself a homology theory for metric spaces, most easily done for finite ones.) But I had no new insights.

We’re in the situation where we know how to take homology for at least two kinds of enriched category:

Graphs, which can be seen as categories enriched in the monoidal category
(ℕ,≥)(\mathbb{N}, \geq), with addition as the monoidal structure. Here we have the Hepworth–Willerton graded homology theory.

Categories themselves. You can define the homology of a (small) category as the homology of its classifying space, or, as for groups, you can do it combinatorially.

Is there a common generalization — a homology theory for enriched categories? I don’t know, and indeed I don’t know whether these two types of homology are similar enough to deserve a common generalization. If there was one, it would give us a homology theory for metric spaces.

But anyway, maybe the step up in generality from graphs to metric spaces is small enough that it’s better to just try to do it directly.

Let me say a little about how magnitude works for graphs. Let XX be a graph, seen as a metric space. The resulting matrix ZXZ_X has entries of the form e−d(x,y)e^{-d(x,y)}. Since distances in graphs are integers, it’s helpful to put q=e−1q = e^{-1} and treat qq as a formal variable. Thus, ZXZ_X is a matrix whose entries are just powers of qq.

This means that the inverse of ZXZ_X is a matrix whose entries are rational functions of qq. Since the magnitude is by definition the sum of the entries of ZX−1Z_X^{-1}, the magnitude is also a rational function of qq. That’s only true for graphs — not metric spaces in general.

You can prove that this rational function — the magnitude — is in fact a power series in qq with integer coefficients. (In principle, you’d only expect it to be a Laurent series with rational coefficients.) Explicitly, it’s

and each xix_i is different from the last: x0≠x1≠⋯≠xkx_0 \neq x_1 \neq \cdots \neq x_k. For example, the coefficient of q0q^0 is the number of vertices and the coefficient of q1q^1 is −2-2 times the number of edges.

For each kk and ℓ\ell, Richard and Simon simply take the free abelian group MCk,ℓ(X)MC_{k, \ell}(X) on the set of such tuples. (“MC” stands for magnitude chain complex.) So we’ve got two gradings here: kk, which is going to be the homological degree, and ℓ\ell, which is going to make this a graded homology theory.

The formula for ∂\partial is on page 7 of Richard and Simon’s paper and I won’t repeat it here. With this in place, we have a chain complex MC•,ℓ(X)MC_{\bullet, \ell}(X) for each ℓ\ell, and its kkth homology is called the kkth magnitude homologyMHk,ℓ(X)MH_{k, \ell}(X) of the graph XX.

The very definition of the group MCk,ℓMC_{k, \ell} — simply the fact that it’s the free abelian group on a certain number of elements — ensures that

Re: A Survey of Magnitude

What stands in the way of a (co)homology of enriched categories?

Simply that we don’t know how to define it. Or I don’t, anyway.

Of course, if we had a good notion of the nerve of an enriched category, and whatever kind of thing that nerve was, it was something we could take the (co)homology of, then we’d be home. But I don’t know of such a construction, and I’m not immediately seeing one in the links you give.

Maybe that’s just my ignorance speaking. If someone else can see a way through, I’d love to know what it is and what (co)homology theory it gives for metric spaces.

I suppose one difficulty is that to cover metric spaces, we need our theory of enriched (co)homology to work for the enriching category V=[0,∞]V = [0, \infty]. Although this is symmetric monoidal closed, and complete and cocomplete, in other ways it’s quite unlike the category of sets, which is everyone’s starting point.

Re: A Survey of Magnitude

Re: A Survey of Magnitude

OK, I’ve read through their construction now. I don’t see how it’s going to help, unfortunately, because their nerve is not the kind of thing whose (co)homology I know how to take.

Their nerve of a 𝒱\mathcal{V}-category 𝒳\mathcal{X} is a certain 𝒱\mathcal{V}-Cat-functor [sic!] D𝒳D_\mathcal{X} from a certain 𝒱\mathcal{V}-Cat-category ℕ\mathbb{N} to SetSet. This ℕ\mathbb{N} has as its objects the natural numbers together with one further object, and depends only on 𝒱\mathcal{V}. It’s not particularly like Δ\Delta or its opposite; in fact, the hom-𝒱\mathcal{V}-category between any two natural numbers is always empty.

Re: A Survey of Magnitude

Thanks David, I’ll have to see if I can figure out what they’re doing.

In my mind, the canonical general framework for cohomology is as follows. You start with a topos ℰ\mathcal{E}. It has associated with it a functor Γ=ℰ(1,−):ℰ→Set\Gamma = \mathcal{E}(1, -) \colon \mathcal{E} \to Set. Now taking abelian group objects throughout, we get an abelian category Ab(ℰ)Ab(\mathcal{E}) and a limit-preserving additive functor Γ:Ab(ℰ)→Ab\Gamma\colon Ab(\mathcal{E}) \to Ab. Assuming there are enough injectives, we can take its right derived functors. The nnth right derived functor of Γ\Gamma probably deserves to be called

Hn(ℰ,−):Ab(ℰ)→Ab.
H^n(\mathcal{E}, -): Ab(\mathcal{E}) \to Ab.

For example:

Let ℰ=Sh(X)\mathcal{E} = Sh(X) for some topological space XX. Then Hn(ℰ,−)H^n(\mathcal{E}, -) would usually be written as Hn(X,−)H^n(X, -). It’s a functor from (sheaves of abelian groups on XX) to AbAb, and it sends a sheaf ℱ\mathcal{F} to Hn(X,ℱ)H^n(X, \mathcal{F}), the nnth cohomology of XX with coefficients in ℱ\mathcal{F}. Of course, we often take ℱ\mathcal{F} to be a constant sheaf.

Let ℰ=Setℂop\mathcal{E} = Set^{\mathbb{C}^{op}} for some small category ℂ\mathbb{C}. Writing Hn(ℰ,−)H^n(\mathcal{E}, -) as Hn(ℂ,−)H^n(\mathbb{C}, -), this means that we can take the cohomology groups of ℂ\mathbb{C} with coefficients in any functor ℂop→Ab\mathbb{C}^{op} \to Ab (that is, in any presheaf of abelian groups on ℂ\mathbb{C}).

In particular, we can take ℂ\mathbb{C} to be a group GG in the last example (as a one-object category). Then we get cohomology groups Hn(G,A)H^n(G, A) for any right GG-module AA. (Again, we often consider the case where AA has trivial GG-action.) This is the usual group cohomology.

I don’t know whether a theory of cohomology for enriched categories could fit into this framework. I don’t know how you’d get a topos, or even an abelian category with a left exact functor on it, from an enriched category.

Also, if we’re trying to generalize Hepworth and Willerton’s homology theory for graphs, we’ll want a grading, and I don’t know where that’s going to come from.

Having a look around for something on enriched toposes and there’s Mike seven years ago:

Enriched topos theory is actually a deep and (at least, to me) unclear subject. So much of topos theory seems to depend on cartesianness. I’ve occasionally thought about what an enriched topos might be, but never really come up with anything really satisfactory. If the enrichment isn’t cartesian monoidal, then the internal logic of an enriched topos would probably be linear logic, but how to interpret linear logic internally in some category is also not an obvious question. One expects that perhaps “quantales” (closed monoidal suplattices) will play a role.

Re: A Survey of Magnitude

Here’s one possible generalization, based on this comment and the ensuing discussion. Let (V,≤,⊗,1)(V,\le,\otimes,1) be a semicartesian monoidal poset, such as ([0,∞],≥,+,0)([0,\infty],\ge,+,0) or (ℕ,≥,+,0)(\mathbb{N},\ge,+,0) or (2={⊥≤⊤},∧,⊤)(\mathbf{2}=\{\bot\le \top\},\wedge,\top). We intend to write down something like a homology theory for finite VV-enriched categories, which in these examples are (related to) metric spaces, graphs, and ordinary posets.

Let CC be a finite VV-category, and let MS(C)MS(C) be the codiscrete simplicial set on the objects of CC, so that MSk(C)=C0k+1MS_k(C) = C_0^{k+1}. This simplicial set is filtered by VV: given any ℓ∈V\ell \in V, define a subset FℓMSk(C)F_\ell MS_k(C) to consist of those kk-simplices (x0,…,xk)(x_0,\dots,x_k) such that there is a map ℓ≤C(xk−1,xk)⊗⋯⊗C(x0,x1)\ell \le C(x_{k-1},x_k) \otimes \cdots \otimes C(x_0,x_1) in VV. The identities 1≤C(x,x)1\le C(x,x) in CC imply that degeneracies preserve the filtration, the composition C(y,z)⊗C(x,y)≤C(x,z)C(y,z) \otimes C(x,y) \le C(x,z) implies that inner faces preserve the filtration, and semicartesianness C(x,y)≤1C(x,y) \le 1 implies that the outer faces do too.

Now if ℓ≤m\ell\le m in VV, then FmMS(C)⊆FℓMS(C)F_m MS(C) \subseteq F_{\ell} MS(C). Thus we can consider the relative homology of this pair of spaces, which can be computed concretely by generating chain complexes of free abelian groups from both of them, taking the quotient chain complex, and then its homology. This gives us a homology theory graded by pairs ℓ≤m\ell\le m (in addition to the usual homological grading kk). We should probably also include the homologies of the individual spaces FℓMS(C)F_\ell MS(C) as part of the theory.

In the case of (ℕ,≥,+,0)(\mathbb{N},\ge,+,0), I think the homology groups Hk,ℓ≤ℓ+1(C)H_{k,\ell\le\ell+1}(C) are the Hepworth-Willerton homology groups. (There is something to say about normalization, but I think that was mostly answered in the previous discussion.) And in the case of 2\mathbf{2}, the space F⊤MS(C)F_\top MS(C) is just the nerve of CC, so the groups Hk,⊤(C)H_{k,\top} (C) are the ordinary poset homology of CC. (Since F⊥MS(C)=MS(C)F_\bot MS(C) = MS(C) is contractible, in this case the other homology groups Hk,⊥≤⊤(C)H_{k,\bot\le\top}(C) carry no additional information.)

Does this give anything interesting for ([0,∞],≥,+,0)([0,\infty],\ge,+,0)? Does it have anything to do with the magnitude?

One could also, of course, imagine various generalizations of this. If VV is a (semicartesian monoidal) category rather than a mere poset, it would be natural to define FℓMSk(C)F_\ell MS_k(C) to consist of kk-simplices (x0,…,xk)(x_0,\dots,x_k)together with a morphism ℓ→C(xk−1,xk)⊗⋯⊗C(x0,x1)\ell \to C(x_{k-1},x_k) \otimes \cdots \otimes C(x_0,x_1). Then it’s no longer a subset of MSk(C)MS_k(C), but that’s not a problem; the associativity and identity axioms for the VV-category CC should ensure that the simplicial identities still hold. The resulting homology theory will be graded by morphisms in VV (and also objects of VV), and when V=SetV=Set it should recover the usual homology of a category. I’m not sure whether we can do without semicartesianness, though.

Re: A Survey of Magnitude

Oh, actually the comparison to Hepworth-Willerton is very direct: the pointed simplicial set Mℓ(G)M_\ell(G) that they define in section 8 of their paper, whose reduced homology is their ℓ\ell-homology of GG, is precisely the cokernel of the map Fℓ−1MS(G)→FℓMS(G)F_{\ell-1} MS(G) \to F_\ell MS(G). Since the latter is a cofibration, its relative homology agrees with the reduced homology of its cokernel.

Re: A Survey of Magnitude

Sorry, other (less interesting) things intervened, so it’s taken me a while to find time to work through this.

First question: am I right in believing that Richard and Simon’s group MHk,ℓ(G)MH_{k, \ell}(G) is, in the notation of your previous comment, Hk,ℓ≥ℓ−1(G)H_{k, \ell \geq \ell - 1}(G)? In that previous comment, you had Hk,ℓ+1≥ℓ(G)H_{k, \ell + 1 \geq \ell}(G) instead.

Re: A Survey of Magnitude

Re: A Survey of Magnitude

Does this give anything interesting for ([0,∞],≥,+,0)([0,\infty],\ge,+,0)?

You’d think we’d be somewhere near persistent homology? I see from p. 3 here that they use a ‘Cech’ (intersecting balls of fixed radius around vertices) and a ‘Rips’ (bounding all edges) simplicial complex. There’s also an alpha complex. But your definition would be about bounding a sum of edges. Might that still give an equivalent homology?

Re: A Survey of Magnitude

Some time ago I mentioned to Simon an old idea of constructing a sort of simplicial complex in which a simplex had a probability attached to it. We thus might have σ\sigma and a P(σ)P(\sigma), measuring the ‘probability of existence of σ\sigma’. If τ\tau is a face σ\sigma, then you would require that the probability of τ\tau existing was greater than that of σ\sigma. (I suspected there would be other conditions but that is the one I recall most clearly.)

The idea was that if we had a sample of measured ‘points’ in a space we could imagine measuring the probability that the ‘point’ was actually at a particular ‘place’, so of the accuracy of the measurement. Think of this as a sort of `fuzz’ around the mean position of the measurement. This gave a measure of how near two points might be, i.e. think of the overlap of two such points, and you have, in the a nerve some sort of 1-simplex with a probability attached to it (as in the paragraph above).

Certainly this idea was not fully baked and my memory of it may be defective, but it may be useful. It did relate to Rips complexes (which are like Vietoris complexes for a sample).

Re: A Survey of Magnitude

The only issue is how to set up the simplicial complex. The Rips approach, working in a symmetric metric space, bounds each of the edges of a simplex by a chosen ϵ\epsilon. You then look to see what persists of the homology for an increase in the parameter ϵ→ϵ′\epsilon \to \epsilon'.

Your approach (working also in the asymmetric case) bounds the sum of nn edges. Might that not lead to greater efficiency, rather than the n⋅(n+1)/2n \cdot (n+1)/2 edges in Rips?

Perhaps there are ‘squeezing’ results for your way in the symmetric case, as on p.6 between Rips and Cech. Any nn-simplex bounded by δ\delta your way is bounded by δ\delta for Rips. Any nn-simplex bounded by ϵ\epsilon the Rips way is bounded by nϵn \epsilon for your way.

Re: A Survey of Magnitude

I see; it does seem very similar, and I agree that there should be a “squeezing” or “interleaving” between these complexes and the Rips ones.

My guess would be that the Rips complex would still be better computationally, though, because as Ghrist says on p3 it’s a “flag complex” determined by its 1-simplices. So although each nn-simplex has n(n+1)/2n(n+1)/2 edges, you don’t have to check each of those edges for each simplex; you can just check all the edges at once, and then the nn-simplies are just those all of whose edges check. Whereas with the magnitude-inspired approach there is actually something to check about each simplex, of arbitrary dimension.

This does mean that this magnitude-inspired homology could be represented by a “barcode”. However, I’m puzzled at why the persistent homology is defined as the image of a homomorphism between homology, rather than as the relative homology of the corresponding map of chain complexes, which seems more natural. Is that also for computational efficiency?

Re: A Survey of Magnitude

The choice of the Rips complex seems to be down to computational ease and the property of “squeezing” the Cech complex. The latter complex is seen as the target because for a given point set in Euclidean space 𝔼n\mathbb{E}^n:

The Cech theorem (or, equivalently, the “nerve theorem”) states that the Cech complex, 𝒞ϵ\mathcal{C}_{\epsilon}, has
the homotopy type of the union of closed radius ϵ/2\epsilon/2 balls about the point. This means that 𝒞\mathcal{C}, though an abstract simplicial complex of potentially high dimension, behaves exactly like a subset of 𝔼n\mathbb{E}^n.

So any feature that persists between the squeezing Rips complexes is there in the squeezed Cech complex.

Re: A Survey of Magnitude

Well, the nn-dimensional homology only depends on the (n+1)(n+1)-skeleton of the simplicial set, so at least the magnitude-inspired HnMSℓH_n MS_\ell sits between the Rips HnRℓ/(n+1)H_n R_{\ell/(n+1)} and HnRℓH_n R_\ell, and similarly in the other direction. Right?

Re: A Survey of Magnitude

Yes, I think that’s right. I was just worrying about the ratio of the squeezing parameters depending on nn, which doesn’t happen for Rips and Cech. But maybe that doesn’t matter.

Anyway, perhaps there are interesting Cech complexes using covers other than equal sized ball forms which are better approximated by the magnitude complex.

Some free association:

There’s a connection between lengths of Hamiltonian paths (our bounds for simplices) and volumes of nn-simplices in the Euclidean case (here). And you can define a simplex volume in a metric space, e.g., here, which is used to determine whether the space can be embedded in ℝn\mathbb{R}^n. So does the volume/path length relationship continue for metric spaces?

Re: A Survey of Magnitude

So is this a graded homology theory for metric spaces (graded by length) that will satisfy Tom as providing that for which his magnitude is a ‘shadow’? Presumably we shouldn’t be taken with the graph version just because it enriches in natural numbers and so gives something comfortingly familiar as a sequence.

(I’ve thought this before. Are they cases where we give something sequence-like greater attention as more readily graspable. Perhaps it was when hearing about RO(G)-grading.)

And what’s going on in the non-posetal case? We don’t usually see that grading “by morphisms in VV (and also objects of VV)”, do we? Is it that in the case of V=SetV = Set that grading isn’t doing much work?

Re: A Survey of Magnitude

Well, it’s totally unclear to me whether it has magnitude as a shadow when applied to metric spaces. It seems that the identification of magnitude as the shadow of Hepworth-Willerton homology relies on an alternative characterization of the magnitude of a graph that isn’t available for general metric spaces. Maybe Tom or someone else who understands magnitude of general metric spaces better can weigh in with an opinion.

Right, in the case V=SetV=Set the grading isn’t doing much work, essentially because because SetSet is freely generated by 11 under coproducts.

Re: A Survey of Magnitude

I was just wondering how likely it was to see examples in the wild of enrichment in the nonposetal cases at semicartesian category and a little Googling brought me back to the Café to the suggestion by Tobias that

the category of stochastic matrices is enriched over convex spaces with tensor product.

Still hard to imagine what the odd indexing in the associated homology theory can do for you (or does something like that coproduct issue arise again?)

Re: A Survey of Magnitude

Note that in the case when VV is cartesian monoidal and also extensive with an indecomposable terminal object, we can identify VV-enriched categories with certain VV-internal categories. In that case, this construction amounts to taking the internal nerve Cat(V)→VΔopCat(V) \to V^{\Delta^{op}} and then applying the Yoneda embedding V→SetVopV \to Set^{V^{op}} levelwise to get a presheaf of simplicial sets on VV.

In particular, therefore, if VV has a small dense subcategory V0V_0, we can use the restricted Yoneda embedding V→SetV0opV \to Set^{V_0^{op}} instead without losing information. This is a fancier way of saying why we see nothing new when V=SetV=Set, because the single object 11 is dense in SetSet, and its restricted Yoneda embedding is the identity.

Another interesting example of this sort is V=GSetV=G Set for a group GG, with V0V_0 the category of orbits G/HG/H. Thus, a category enriched over GG-sets has a homology indexed by orbits, which I think is just the usual equivariant (Bredon) homology of its GG-equivariant nerve. (Curiously, I don’t think I recall ever seeing the “morphism-graded” version in equivariant homotopy theory. Maybe it’s present implicitly somewhere?)

Re: A Survey of Magnitude

Hi both,

Thanks for all these comments! I spent the last few days mostly unplugged from the internet, and now I want to spend a while reading and thinking about what you’ve written.

As I wrote back here, my interest in a hypothetical magnitude homology for metric spaces was rekindled by going to a conference on applied topology and sitting in talks on persistent homology. It’s easy to fantasize that the two might be connected.

More broadly, it would be great to give a treatment of persistent homology in the spirit of Lawvere’s article “Taking categories seriously” — that is, as an inevitable outgrowth of the observation that metric spaces are enriched categories. I don’t know whether that quest is completely separate from the quest for a magnitude homology theory for metric spaces, or essentially the same as it, or somewhere in between.

Re: A Survey of Magnitude

Quick question: how much of the theory of magnitude of graphs (including Hepworth-Willerton homology) depends on having a graph, and how much of it depends only on having a metric space whose distances are all integers?

Re: A Survey of Magnitude

I need to pick up my folder of notes from the work Aaron and I did, which is currently in a different city from me. Then I may be able to give a more precise answer. But in the meantime, I have it in my head that among all metric spaces, graphs are special in two ways.

The first is that the distances are integers. The second is that in a suitably discrete space, graphs are geodesic — or if you prefer, “convex”.

By definition, a metric space XX is geodesic if for any two points xx and yy a finite distance DD apart, there is a distance-decreasing map γ:[0,D]→X\gamma\colon [0, D] \to X such that γ(0)=x\gamma(0) = x and γ(D)=y\gamma(D) = y. For instance, a closed subset of ℝn\mathbb{R}^n is geodesic if and only if it is convex.

There is a similar concept for metric spaces with integer distances. Such a space XX could be said to be discretely geodesic if for any two points xx and yy a finite distance DD apart, there is a distance-decreasing map γ:[[0,D]]→X\gamma\colon [[0, D]] \to X such that γ(0)=x\gamma(0) = x and γ(D)=y\gamma(D) = y. Here [[0,D]][[0, D]] is impromptu notation for the graph

0 — 1 — ⋯\cdots — D.

Categorically, [0,D][0, D] and [[0,D]][[0, D]] are something like the coslice categories D/[0,∞]D/[0, \infty] and D/ℕD/\mathbb{N}, respectively. I say “something like” because there are symmetry issues to be considered.

Anyway, the point is that all graphs are discretely geodesic, simply because that’s how the metric is defined. So that’s their second special property.

Re: A Survey of Magnitude

Here’s another approach to categorifying magnitude, which seems to more or less fail. But I had fun thinking about it.

Tom didn’t say much about it in this post, but the magnitude of a metric space is a special case of the magnitude of an enriched category. The latter is defined relative to any monoidal category VV equipped with a monoid homomorphism |⋅||\cdot | from π0(V)\pi_0(V), its monoid of isomorphism classes of objects, to the multiplicative monoid of some rig kk. The magnitude of a VV-category is then (when defined) an element of kk: we apply |⋅||\cdot | to the hom-objects to get a kk-valued matrix, then add up the entries of any weighting or coweighting if it has both (or the elements of its inverse, if it has one).

Of course there is a universal choice of kk, namely the “monoid rig” (the inverse-free version of a “group ring”) of π0(V)\pi_0(V), whose elements are ℕ\mathbb{N}-linear combinations of elements of π0(V)\pi_0(V). (For instance, if V=(ℕ,≥)V = (\mathbb{N},\ge) for “metric spaces with integer distances” including graphs, then this monoid rig is the rig of polynomials in one variable with natural number coefficients.) Note that if we define magnitude using this universal choice of kk, then it will be a finite ℕ\mathbb{N}-linear combination of isomorphism classes of objects of VV, or equivalently a finitely-supported function from π0(V)\pi_0(V) to ℕ\mathbb{N}. Thus, the magnitude is naturally already “graded by objects of VV”, so it makes sense that a categorification of it would be too.

Now, if we want magnitude to be the “Euler characteristic” of a homology theory, we ought to be over a ring rather than a rig, since Euler characteristic involves (−1)k(-1)^k. Thus, let’s consider instead the “monoid ring” ℤ[π0(V)]\mathbb{Z}[\pi_0(V)], which for simplicity we may write ℤ[V]\mathbb{Z}[V].

If we also want to generalize some of Tom’s arguments about graph magnitude, we need an analogue of “power series”. For this purpose, suppose that VV contains no nontrivial invertible objects; that is, if X⊗Y≅IX\otimes Y \cong I, then X≅IX\cong I and Y≅IY\cong I, where II is the unit object. Then the non-unit isomorphism classes generate a proper ideal ℐ\mathcal{I} in ℤ[V]\mathbb{Z}[V]. Let kk be the completion of ℤ[V]\mathbb{Z}[V] at that ideal. Informally, in kk we are allowed to formally add up infinite ℤ\mathbb{Z}-linear combinations of objects of VV as long as each of those objects is decomposed as a nontrivial (i.e. not involving the unit object) nn-ary tensor product in such a way that each nn appears only finitely many times.

Now we can define magnitude of VV-categories relative to kk. It seems that we should then be able to reproduce the proof of Proposition 3.9 from here, leading to a similar formula for the magnitude of a VV-category in terms of tuples of objects with adjacent ones distinct for which the tensor product of their hom-objects belongs to a particular isomorphism class. One could then try to mimic the Hepworth-Willerton construction on such tuples, and get a “categorification” of magnitude for cardinality reasons as Tom mentioned.

However, this doesn’t help for metric spaces, because the completion can actually destroys a lot of information. In particular, if a single object of VV can be decomposed as a nontrivial nn-ary tensor product for all nn, then it maps to 00 in kk. (Formally this is because it lies in ⋂nℐn\bigcap_n \mathcal{I}^n; informally this property would mean we could add up infinitely many copies of it, leading to an Eilenberg swindle.) This is unfortunate because x=xn+⋯+xnx = \frac{x}{n} + \cdots + \frac{x}{n} in [0,∞][0,\infty] for any xx, so that when V=[0,∞]V=[0,\infty] we have k=ℤk=\mathbb{Z} with all nonzero x∈[0,∞]x\in[0,\infty] mapping to 0∈ℤ0\in\mathbb{Z}. So it seems that this approach isn’t going to tell us anything about the magnitude of metric spaces.

Re: A Survey of Magnitude

Is there any way to generalize the Hepworth–Willerton homology from graphs to general finite metric spaces?

and Mike’s suggestion for how to do so, here is a basic calculation that seems likely to be central to making progress. It’s going to be a purely formal calculation, done without regard to questions of convergence etc.

Write Z=(Z(i,j))Z = (Z(i, j)). If all the diagonal entries of ZZ are 11 (as is the case when ZZ is the matrix coming from a metric space or graph or poset), then we can rewrite this expression for the magnitude of ZZ as

This set 𝕃A⊆[0,∞]\mathbb{L}_A \subseteq [0, \infty] is not in general finite, but it’s not too hard to see that 𝕃A∩[0,L]\mathbb{L}_A \cap [0, L]is finite for all real L≥0L \geq 0. So summing over 𝕃A\mathbb{L}_A is very like summing over ℕ\mathbb{N} — in other words, entirely manageable.

Re: A Survey of Magnitude

Thanks for writing that out, Tom! I had something similar going around in my head, but couldn’t see it as clearly as that.

I think that for ℓ∈𝕃A\ell\in\mathbb{L}_A, your ck,ℓc_{k,\ell} is the rank of my chain group MSk,ℓ≥ℓ−εMS_{k,\ell\ge \ell-\varepsilon}, where ε\varepsilon is small enough that (ℓ−ε,ℓ]∩𝕃A={ℓ}(\ell-\varepsilon,\ell] \cap \mathbb{L}_A = \{\ell\}. Thus ∑k≥0(−1)kck,ℓ\sum_{k\ge 0} (-1)^k c_{k,\ell} is the Euler characteristic of the chain complex MSℓ≥ℓ−εMS_{\ell\ge \ell-\varepsilon}, and hence also of the homology MH*,ℓ≥ℓ−εMH_{\ast,\ell\ge\ell-\varepsilon}. So insofar as your formal calculation is meaningful, it should mean that the magnitude of a finite metric space is indeed an Euler characteristic of my proposed homology theory.

The other thought I had is that maybe we could make this formal calculation meaningful by approximating a finite metric space by ones we understand better. For instance, it seems to me that at least in the “generic” case when the triangle inequality is never an equality (i.e. there are no “collinear triples”), we should be able to deform a finite metric space slightly to make its distances all integer multiples of some fixed small positive real number, and that the formal-power-series calculation that works for graphs ought also to work for metric spaces with that property. Does that seem plausible?

Re: A Survey of Magnitude

Yes, absolutely. Reflecting on your original comment, I had in my head the rather melodramatic notation limε→0+MH*,ℓ≥ℓ−ε\lim_{\varepsilon \to 0+} MH_{\ast, \ell \geq \ell - \varepsilon}. Perhaps it wouldn’t be too misleading to write
MH*,ℓ−MH_{\ast, \ell-}, much as people sometimes write f(x−)f(x-) to mean limε→0+f(x−ε)\lim_{\varepsilon \to 0+} f(x - \varepsilon).

(I can’t help mentioning a possible connection with something else, though maybe it’s a red herring. In a post about the Euler calculus — Integrating against the Euler characteristic — I mentioned the formula ∫fdχ=∑x∈ℝ(f(x)−f(x−))\int f \,d\chi = \sum_{x \in \mathbb{R}} (f(x) - f(x-)) for the integral against the Euler characteristic of a function ff on ℝ\mathbb{R}. I think Euler characteristic is playing a different role there, and I can’t see any definite relevance of this formula, but neither can I resist mentioning it.)

Regarding your last paragraph, I see what you mean. The hypothesis that all distances are integer multiples of some fixed real was something Aaron and I occasionally adopted. Incidentally, one crucial thing we didn’t do, which you did, was think about relative homology.

One potential problem, depending on what one is going to do with this integer multiples idea, is that we’re short on stability theorems for magnitude. For instance, we don’t know whether the magnitude of finite subsets of ℝn\mathbb{R}^n is continuous with respect to the Hausdorff metric. It’s fairly clear that it’s continuous if you keep the number of points the same, but it’s not so clear when the number of points changes.

Re: A Survey of Magnitude

I think that for ℓ∈𝕃A\ell \in \mathbb{L}_A, your ck,ℓc_{k, \ell} is the rank of my chain group MSk,ℓ≥ℓ−εMS_{k, \ell \geq \ell - \varepsilon}

I initially nodded along to this, because I’d been through some similar thought process, but now I’m confused.

We have a simplicial set MSMS, and a filtration (FℓMS)ℓ≥0(F_\ell MS)_{\ell \geq 0} on it. I’d be happy to write MSk,ℓMS_{k, \ell} for FℓMSkF_\ell MS_k, and in fact I’ve been doing exactly that in private. We can form the free abelian group on it, ℤMSk,ℓ\mathbb{Z} MS_{k, \ell}, and that gives a chain complex ℤMS*,ℓ\mathbb{Z} MS_{\ast, \ell} for each ℓ\ell. It has a subcomplex ℤMS*,ℓ−ε\mathbb{Z} MS_{\ast, \ell - \varepsilon}, and we can form the cokernel of the inclusion:

Re: A Survey of Magnitude

Yes, I was being sloppy with the notation, and also glossing over the passage to the normalized chain complex which I think deals with the issue of nondegeneracy. Let me reply more carefully to your comment below.

Re: A Survey of Magnitude

One potential problem, depending on what one is going to do with this integer multiples idea, is that we’re short on stability theorems for magnitude. For instance, we don’t know whether the magnitude of finite subsets of ℝn\mathbb{R}^n is continuous with respect to the Hausdorff metric. It’s fairly clear that it’s continuous if you keep the number of points the same, but it’s not so clear when the number of points changes.

What I had in mind would be keeping the number of points the same. If all the triangle inequalities are strict, then there is a minimum nonzero value of d(x,y)+d(y,z)−d(x,z)d(x,y) + d(y,z) - d(x,z), say η\eta. So if we modify all distances by less than η/3\eta/3, the triangle inequalities will still all hold. Now for any ε<η/3\varepsilon \lt \eta/3, let dε(x,y)d_\varepsilon(x,y) be the integer multiple of ε\varepsilon closest to d(x,y)d(x,y).

I feel like it should be possible to deal with triangle equalities as well, since in that case we essentially have to simultaneously approximate a finite number of (potentially) irrational numbers by rational ones, but I’m not sure exactly how to write it out.

Re: A Survey of Magnitude

The goal of the integer multiples idea was to make Tom’s “formal calculation” rigorous by approximating a metric space by a space where the objects in question can be regarded as actual formal power series.

Re: A Survey of Magnitude

Doesn’t this make point 4 of the post clear?

So the limit of H*,=ℓ(tA)H_{\ast,= \ell}(t A) as t→∞t \to \infty is the limit of H*,=ℓ(A)H_{\ast,= \ell}(A) as ℓ→0\ell \to 0. For small enough ℓ\ell, the complex will just involve the 0-simplexes of the points in the point set.

Re: A Survey of Magnitude

That’s a nice argument! But I have to confess, item 4 — the fact that limt→∞|tA|=card(A)\lim_{t \to \infty} |t A| = card(A) — probably shouldn’t have been on that “list of things we used not to know and do know”, since it was actually worked out very early on.

One way to see it is purely topological. When tt is large, the matrix ZtAZ_{t A} of tAt A is close to the n×nn \times n identity matrix II, where n=card(A)n = card(A). In a neighbourhood of II, all matrices are invertible, and the sum of the entries of the inverse matrix varies continuously. Since the sum of the the entries of the inverse of II is nn, that implies that limt→∞|tA|=n\lim_{t \to \infty} |t A| = n.

Re: A Survey of Magnitude

If you can do that, David, it would be spectacular!

Item 5 says that for compact A⊆ℝnA \subseteq \mathbb{R}^n,
Vol(A)=cnlimt→∞|tA|tn
Vol(A) = c_n lim_{t \to \infty} \frac{|tA|}{t^n}
where cnc_n is a known constant. It was proved as Theorem 1 of this paper by Barceló and Carbery using Fourier analysis. (The numbering in that paper is a bit eccentric; Theorem 1 is stated on page 8 and its proof is on pages 10–12.) They call it “elementary Fourier analysis”, but you can see that it’s a page and a half of fairly serious calculation involving Sobolev spaces and such.

Item 6 says that for compact A⊆ℝnA \subseteq \mathbb{R}^n, the Minkowski dimension of AA is equal to the growth of the magnitude function. This was proved by Mark Meckes (Cor 7.4), again using a significant amount of analysis, and also using the notion of maximum diversity.

To find a mainly-homological proof of either of these would be a real advance!

Re: A Survey of Magnitude

It may be worth noting that my proposed generalization of magnitude homology to metric spaces doesn’t depend on finiteness: just like every small category has a homology, so does every metric space. One might therefore fantasize that perhaps the magnitude of a compact metric space could be extracted directly from its homology.

Re: A Survey of Magnitude

Yes, sorry — that should have been clear. I’m no longer sure why I mentioned the problems that arise when varying the number of points. Maybe it was just an excess of caution.

I’m with you on both perturbation ideas (the rational-multiple one and the no-collinearity one). At the same time, I’m somehow not too concerned with the question of convergence of infinite expressions such as in this comment. For some reason, I have faith that it will work out if we take care of everything else properly.

Re: A Survey of Magnitude

Write sum(M)sum(M) for the sum of all the entries of a matrix MM. I believe the following is true.

Theorem Let ZZ be an invertible matrix over a field (or maybe any commutative ring). Then:

The power series fZ(t)=∑k=0∞sum((Z−I)k)tkf_Z(t) = \sum_{k = 0}^\infty sum((Z - I)^k) t^k is a rational function.

fZf_Z does not have a pole at −1-1.

fZ(−1)=|Z|f_Z(-1) = |Z|.

In other words, we sum the divergent series ∑k(−1)ksum((Z−I)k)\sum_k (-1)^k sum((Z - I)^k) by changing −1-1 to a formal variable tt, noting that the formal power series is formally equal to a rational function, and evaluating that function at −1-1.

I think parts 2 and 3 are also true. It’s basically Theorem 3.2 of that paper, but I’d need to go back through it in detail in order to be sure.

(Somewhat technical footnote follows.)

Actually, Theorem 3.2 suggests something a bit more general. If you ask whether the matrix ZZ of an (enriched) category is invertible, the answer isn’t invariant under equivalence. In fact, if a category isn’t skeletal then its matrix has two identical rows and is therefore not invertible. So the result above looks a bit uncategorical.

Let’s say that a square matrix is essentially invertible if it can be obtained from an invertible matrix by repeatedly duplicating rows and columns, but always duplicating the same rows as columns. For instance, we might start with an invertible 6×66 \times 6 matrix, insert a new 2122\frac{1}{2}th row identical to the existing 4th row, and insert a new 2122\frac{1}{2}th column identical to the existing 4th column. The resulting 7×77 \times 7 matrix would be essentially invertible.

The point is that if a category is equivalent to one whose matrix is invertible, then its own matrix is essentially invertible. And I think the “Theorem” stated above is true for essentially invertible matrices, not just invertible ones. That would be a more categorically relevant result.

Re: A Survey of Magnitude

That’s nice! So then all that’s left to worry about would be the sum over 𝕃A\mathbb{L}_A in the rearranged sum.

Actually, it seems like this argument should work to compare Euler characteristic with magnitude homology for arbitrary semicartesian-enriched categories, with e−ℓe^{-\ell} replaced by a suitable monoid homomorphism into the multiplicative monoid of some ring? I guess in the general case 𝕃A\mathbb{L}_A is likely to be even more problematic.

Glancing over the paper you just linked to makes me wonder whether magnitude homology is invariant under equivalence of enriched categories, since series Euler characteristic is not. But the ordinary homology of an ordinary category is so invariant, so I’m confused.

Re: A Survey of Magnitude

Re invariance, I very much hope that magnitude homology is equivalence-invariant. It would be alarming if not!

The failure of series Euler characteristic to be invariant only happens when you step outside the realm of categories that “essentially have Möbius inversion”. (By definition, a category AA has [crude] Möbius inversion if its matrix ZAZ_A is invertible, and it essentially has Möbius inversion if it’s equivalent to one that actually does.) Within that realm, it is invariant under equivalence, because it agrees with ordinary Euler characteristic. I think that resolves the apparent contradiction…?

Re: A Survey of Magnitude

Maybe it should resolve it, but I’m still confused, because magnitude homology looks to me more like series Euler characteristic than like ordinary Euler characteristic (where I assume by “ordinary” you mean the weighting-coweighting definition for categories, not the classical topological definition). So why should I be reassured by the invariance of series Euler characteristic for categories with essential Möbius inversion, since magnitude homology is defined for all categories?

Re: A Survey of Magnitude

I assume by “ordinary” [Euler characteristic] you mean the weighting-coweighting definition for categories, not the classical topological definition

Yes, that’s what I meant. But I suspect you know that they agree, under finiteness hypotheses, in the sense that the Euler characteristic of a category (defined by weightings/coweightings) is equal to the topological Euler characteristic of its nerve or geometric realization.

I didn’t have anything sophisticated in mind when I made my comment that I didn’t think there was any contradiction. The theorem about getting |Z||Z| as the sum of a divergent series relies on ZZ being invertible (or at least essentially invertible). If we’re imposing that hypothesis, then we’re in a context where series Euler characteristic is invariant under equivalence. If we drop that hypothesis then we lose the invariance of series Euler characteristic — but we also lose the link between magnitude homology and Euler characteristic, since the theorem fails. So, the fact that series EC isn’t invariant doesn’t imply that magnitude homology isn’t. That’s all I had in mind.

Re: A Survey of Magnitude

I suspect you know that they agree, under finiteness hypotheses

Yes, I do. In fact, Kate and I recently re-proved that the weighting Euler characteristic coincides with the general duality-theoretic notion of Euler characteristic in section 7 of this paper. But the finiteness hypotheses are, if I recall correctly, fairly restrictive, and in particular rule out all non-integral Euler characteristics (as they must, since topological Euler characteristics are always integers). Similarly, here I was worried about the non-homotopy-invariance of series Euler characteristic, which also seems to happen only in the realm where it goes beyond topological Euler characteristic. Right?

Re: A Survey of Magnitude

Ah yes, I looked over your paper with Kate when it appeared in TAC, but I’m not sure I saw section 7. Did you speak on some part of the content of that paper when you came to Glasgow?

The finiteness hypotheses I was referring to were that the nerve of the category has only finitely many nondegenerate simplices, or equivalently that the category is finite, skeletal, and contains no endomorphisms other than identities.

When that’s the case, the matrix of the category is invertible over ℚ\mathbb{Q} (Theorem 1.4 of The Euler characteristic of a category). That in turn implies that series Euler characteristic agrees with ordinary (i.e. “my”) EC. Ordinary EC is invariant under equivalence, and in fact two categories whose EC is defined have the same EC if there exists an adjunction between them.

So: under these strong finiteness hypotheses, series EC of categories is indeed “homotopy invariant”, whether that’s interpreted to mean invariance under equivalence or in the stronger sense of invariance under the existence of an adjunction. Or more briefly:

Re: A Survey of Magnitude

My talk in Glasgow was kind of a conglomeration of all my most recent papers with Kate. Euler characteristics of categories popped up finally on the penultimate slide.

One of my idle dreams is to find an (∞\infty-)category in which those strong finiteness restrictions can be weakened, so as to include at least finite groupoids, yet retain the identification of your Euler characteristic with a symmetric-monoidal trace. But I’ve tried various things and nothing seems to work.

Re: A Survey of Magnitude

Suppose I have a simplicial set XX with finitely many simplices in each dimension. Then I can define three formal power series:

fX(t)=∑n≥0cntn f_X(t) = \sum_{n\ge 0} c_n t^n

gX(t)=∑n≥0dntn g_X(t) = \sum_{n\ge 0} d_n t^n

hX(t)=∑n≥0bntn h_X(t) = \sum_{n\ge 0} b_n t^n

where cnc_n is the number of nondegenerate nn-simplices, dnd_n is the total number of nn-simplices (degenerate or non), and bnb_n is the rank of Hn(X)H_n(X) (the Betti numbers of XX). Note that fXf_X and gXg_X are not invariant under weak homotopy equivalence of XX, whereas hXh_X is. Of course, fXf_X and gXg_X are the generating functions of the ranks of the chain complexes C*nd(X)C_\ast^{nd}(X) and C*(X)C_\ast(X), while hnh_n is the generating function of the ranks of the homology H*(X)H_\ast(X).

IFXX has finitely many nondegenerate simplices in total, then fXf_X and hXh_X are both simply polynomials, and a standard argument shows fX(−1)=hX(−1)f_X(-1) = h_X(-1), which is the Euler characteristic of XX. In particular, in this case fX(−1)f_X(-1) does happen to be homotopy-invariant, because it’s equal to hX(−1)h_X(-1).

However, if XX is not finite-dimensional, we could consider analytically continuing any one of the above three power series to −1-1. In general there is no reason for the results to agree, and thus no reason for fX(−1)f_X(-1) (or gX(−1)g_X(-1)) to be homotopy-invariant. Your definition of “series Euler characteristic” uses fX(−1)f_X(-1), and your counterexample shows that this is in fact not homotopy-invariant — and therefore is not always equal to hX(−1)h_X(-1), since the latter must be homotopy-invariant as it’s defined in terms of homotopy-invariant homology.

So this doesn’t give any reason to worry that magnitude homology might not be invariant under equivalence. (It would still be good to have a proof that it is, of course.) But it does make me wonder how helpful the analytic-continuation approach will be in relating magnitude homology to magnitude, since the formal calculation with divergent series uses an analogue of fXf_X rather than hXh_X. How is this dealt with in the case of graphs?

Re: A Survey of Magnitude

Maybe I should think a little longer before posting. I guess what’s happening for graphs is that after we rearrange the series to sum over ℓ\ell first, then the individual sums ∑n≥0(−1)ncn,ℓ\sum_{n\ge 0} (-1)^n c_{n,\ell} are finite and correspond to finite-dimensional chain complexes MC=ℓndMC^{nd}_{=\ell}: since all distances between distinct vertices are positive integers, there can’t be any nondegenerate nn-simplices of length ℓ\ell if n>ℓn\gt \ell. The same thing should work for finite metric spaces as long as they are skeletal (d(x,y)=0⇒x=yd(x,y)=0 \Rightarrow x=y) since then there is a minimum distance between any two distinct points.

I guess there’s still the question of whether the rearrangement is valid, though.

Re: A Survey of Magnitude

Right — for graphs, Richard and Simon repeatedly exploit the fact that MCk,ℓ=0MC_{k, \ell} = 0 for k>ℓk \gt \ell. So, the same is true of MHMH. Everything’s “below the diagonal” (at least, the way they write it!). They’re particularly interested in the case where the magnitude homology is only nonzero on the diagonal, which happens quite often.

As you say, one consequence of the fact that MCk,ℓ=0MC_{k, \ell} = 0 when k>ℓk \gt \ell is that for each ℓ\ell, the in-principle-infinite sum
∑k=0∞(−1)krank(MHk,ℓ)
\sum_{k = 0}^\infty (-1)^k rank(MH_{k, \ell})
is in fact finite.

Re: A Survey of Magnitude

Fix a graph, which in this comment will remain nameless. Also take a natural number kk (to be thought of as a
homological degree) and a natural number ℓ\ell (to be thought of as a
length). From this data, there are several closely related sets we can
define:

Here “nd” stands for “nondegenerate”, and in all cases x0,…,xkx_0, \ldots, x_k
are vertices of the graph. As you see, I’m trying to be
systematic with the notation, at the risk of making things worse.

For a fixed ℓ\ell and varying kk, all four are semisimplicial sets (with
face maps but not degeneracies in all cases). For the two with a
“≤ℓ\leq\ell” subscript, the formula for the face maps is the standard thing
(omit one of the arguments). For the two with an “=ℓ=\ell” subscript, the
formula for the face maps involves two cases, as in Definition 2 of
Hepworth–Willerton.

We can form four chain complexes from these four semisimplicial sets, by
the usual recipe of taking free abelian groups and then alternating sums of
face maps. Call them C*,=ℓC_{\ast, = \ell}, C*,=ℓndC^{nd}_{\ast, =\ell}, C*,≤ℓC_{\ast,
\leq\ell} and C*,≤ℓndC^{nd}_{\ast, \leq\ell}. Comments on each of them:

C*,=ℓC_{\ast, =\ell} is the chain complex coming from Richard and Simon’s
simplicial set MℓM_\ell (Definition 40 of their paper and mentioned by
Mike here), give or take a
basepoint.

C*,=ℓndC^{nd}_{\ast, =\ell} is Richard and Simon’s magnitude chain
complex MC*,ℓMC_{\ast,\ell}. Its homology is, by definition, the magnitude homology of the graph.

C*,≤ℓC_{\ast, \leq\ell} is the chain complex coming from Mike’s simplicial
set FℓMSF_\ell MS.

I don’t think we’ve seen C*,≤ℓndC^{nd}_{\ast, \leq\ell} before.

Now I want to know how these four chain complexes relate to one another.

Question Do C*,=ℓC_{\ast, =\ell} and C*,=ℓndC^{nd}_{\ast, =\ell} have
the same homology? What about C*,≤ℓC_{\ast, \leq\ell} and C*,≤ℓndC^{nd}_{\ast,
\leq\ell}?

This is the question that I imagine is easy to answer for anyone
competent in homological algebra.

Here’s a different relationship between these four chain complexes that I
think I do understand. There are short exact sequences of chain complexes

In both cases, the first map is the inclusion and the second annihilates
all tuples of length <ℓ\lt \ell.

Now, one difference between the complexes with an “nd” and those without is
that the “nd” ones are bounded. (This is simply because if x0≠⋯≠xkx_0 \neq \cdots
\neq x_k then d(x0,x1)+⋯+d(xk−1,xk)≥kd(x_0, x_1) + \cdots + d(x_{k - 1}, x_k) \geq k.) The boundedness allows us to
do some easy calculations with Euler characteristic, as
follows.

Generally, given a short exact sequence 0→A→B→C→00 \to A \to B \to C \to 0 of
bounded chain complexes, we get a homology long exact sequence which is
bounded and therefore has Euler characteristic 00. From this it
follows that χ(H*(A))−χ(H*(B))+χ(H*(C))=0\chi(H_\ast(A)) - \chi(H_\ast(B)) + \chi(H_\ast(C)) = 0. So
in the case at hand, writing H*,=ℓndH^{nd}_{\ast, =\ell} for the homology of
C*,=ℓndC^{nd}_{\ast, =\ell} etc., we have

I’m not sure what to make of this. The left-hand side is the
magnitude of the graph. The right-hand side isn’t in principle the same as
the Euler characteristic that Mike was calculating, because of the “nd”,
although maybe it actually is the same. Does this equation get us anywhere?

Re: A Survey of Magnitude

You probably won’t be surprised to hear that I think our attention should focus on Sk,≤ℓS_{k,\le\ell}; everything else can be understood in relation to it. Note that it is actually a simplicial set, not just a semisimplicial one: if we don’t demand nondegeneracy, then we can duplicate points to give degeneracies.

On the other hand, I don’t actually believe that any of the other three are even semisimplicial sets until you either add a basepoint or take free abelian groups. Definition 2 of Hepworth-Willerton uses “0” in the second case, which doesn’t exist until you do one of those things. They don’t say explicitly what happens if x0=x2x_0 = x_2 and your face map is omitting x1x_1, but since in that case the distance would no longer be ℓ\ell it’s also covered by the second case.

So actually,all we have to start with is one family of (unpointed) simplicial sets S*,≤ℓS_{\ast,\le\ell}, which gives rise by free abelian groups to a simplicial abelian group ℤS*,≤ℓ\mathbb{Z} S_{\ast,\le\ell}. Now there are several ways to make a chain complex out of a simplicial abelian group, which are summarized here. One is the “alternating sum of face maps” (which only sees the underlying semisimplicial structure), which gives your C*,≤ℓC_{\ast,\le\ell}. Another is the quotient of this by all the degenerate simplicies, which is probably what you have in mind for C*,≤ℓndC_{\ast,\le\ell}^{nd}.

The important theorem is that this quotient map induces an isomorphism on homology (and, of course, if we started with the free simplicial abelian group on a simplicial set, their common homology is the homology of that simplicial set in the topological sense). So for purposes of homology, it doesn’t matter whether you quotient out the degeneracies first, and so in particular C*,≤ℓC_{\ast,\le\ell} and C*,≤ℓndC_{\ast,\le\ell}^{nd} have the same homology.

Now, we have an inclusion S*,≤ℓ−1→S*,≤ℓS_{\ast,\le\ell-1} \to S_{\ast,\le\ell} of simplicial sets. The quotient/cofiber of this inclusion, meaning the pushout of 1←S*,≤ℓ−1→S*,≤ℓ 1\leftarrow S_{\ast,\le\ell-1} \to S_{\ast,\le\ell}, is your “S*,=ℓS_{\ast,=\ell}” but with a basepoint added (the image of 11) to be the image of face maps that lead out of “(=ℓ)(=\ell)-land”. As you essentially said, this is isomorphic to Richard and Simon’s MℓM_\ell.

Now if you have a pointed simplicial set, the natural way to make it a simplicial abelian group is to take the free abelian groups on the simplices but set the basepoint simplices to 00. The homology of this is what topologists call the “reduced” homology of the pointed simplicial set you started with, and can be obtained by either the “alternating sum” chain complex or the version that quotients out the degeneracies first. Applied to S*,=ℓS_{\ast,=\ell} these two constructions should give what you have in mind for C*,=ℓC_{\ast,=\ell} and C*,=ℓndC_{\ast,=\ell}^{nd} respectively. Thus, they also have the same homology, which is the magnitude homology of the graph.

The short exact sequences also arise from this construction. Given any map of unpointed simplicial sets X→YX\to Y, its cofiber (pushout of 1←X→Y1 \leftarrow X \to Y) is, as I mentioned above, canonically pointed by the image of 11. On the other hand, we can add disjoint basepoints to XX and YY, obtaining a map of pointed simplicial sets X+→Y+X_+ \to Y_+, and then the pushout 1←X+→Y+1 \leftarrow X_+ \to Y_+ in the category of pointed simplicial sets is once again the same cofiber. Since the “free abelian group on a pointed set” (which sets the basepoint to 00) is a left adjoint, and takes 11 to 00, applying it levelwise to this we obtain a short exact sequence of simplicial abelian groups. Passing to chain complexes in the two ways discussed above, we get your two short exact sequences of chain complexes.

However, I wasn’t proposing to calculate the Euler characteristic of H*,≤ℓH_{\ast,\le\ell}, but rather that of H*,=ℓH_{\ast,=\ell} to get exactly the magnitude homology, by noting that in the case of a graph (or more generally of integer distances) m=ℓm=\ell means the same as ℓ−1<m≤ℓ\ell-1 \lt m \le \ell.

Re: A Survey of Magnitude

So, reusing similar notation in the more general case of a finite metric space, your ck,ℓc_{k,\ell} is the rank of Ck,ℓ−ndC_{k,\ell^-}^{nd}, and hence ∑k≥0(−1)kck,ℓ\sum_{k\ge 0} (-1)^k c_{k,\ell} is the Euler characteristic of C*,ℓ−ndC_{\ast,\ell^-}^{nd}, which is the same as that of its homology, which is also the homology of C*,ℓ−C_{\ast,\ell^-}.

Re: A Survey of Magnitude

Thanks Mike; that’s enormously helpful.

Of course you’re right about various things not even being semisimplicial. I was really thinking to myself “we have some maps ∂i\partial_i and if we take their alternating sums then we get a chain complex”. From there I lazily leapt to the conclusion that we had a semisimplicial set, forgetting that the definition of those maps ∂i\partial_i required something called 00. This makes me feel bad, as I seem to remember bugging Aaron on that point.

You probably won’t be surprised to hear that I think our attention should focus on Sk,≤ℓS_{k, \leq\ell}

Absolutely. It’s clearly the categorically natural thing, so it should play the primary role.

Nevertheless, at some point we’ll need the nondegenerate and =ℓ=\ell variants in order to make the connection with Richard and Simon’s work.

However, on this thread we’ve neglected to mention something obvious. In many finite metric spaces, the triangle inequality is strict — I mean, equality never holds except in the trivial case where the first two or last two points are the same. Indeed, the triangle inequality is strict in almost all finite metric spaces, in some sense that I won’t try to make precise. For example, if you choose “at random” a finite point-set in ℝn\mathbb{R}^n then there’s probability 00 that three of the points lie on a straight line, hence probability 11 that the triangle inequality in it is strict.

What that means is that in most finite metric spaces, the differential ∂\partial is 00. So, the magnitude homology groups are simply the chain groups. This doesn’t mean anything’s gone wrong, but it does mean that you don’t see the kind of reduction that you often see in other homology theories, wherein the chain groups are huge but the homology groups aren’t.

Earlier in this thread, I pointed out two special properties of graphs as metric spaces. The more obvious one is that distances are integers. The less obvious one is that the distances in graphs are geodesic. So in a graph, equality happens quite often in the triangle inequality. That means that you do tend to get a real reduction in size when you pass from chains to homology, as Richard and Simon point out on page 3 of their paper.

When you look at geometrically interesting infinite metric spaces, you’re usually going to get equality happening quite often in the triangle inequality. For instance, it happens all over the place in convex sets (which might be thought of as analogous to graphs, since distances in them are geodesic). On the other hand, there are still geometrically interesting infinite metric spaces where the triangle inequality is strict, such as the circle with the subspace metric from ℝ2\mathbb{R}^2.

Re: A Survey of Magnitude

That’s a good point! My immediate reaction is that it means that magnitude homology of metric spaces will be most interesting when we start using coefficients other than the AℓA_\ell’s that allow us to exactly reproduce the Hahn series representation of magnitude.

In particular, the ℓ\ell-as-coefficients perspective doesn’t force us to grade the homology groups by a pair of values ℓ1≥ℓ2\ell_1\ge \ell_2, but we are certainly free to do so if we want; indeed for any interval I⊆[0,∞]I\subseteq [0,\infty] (open, closed, or half-open) we can define AI(ℓ)=ℤA_I(\ell) = \mathbb{Z} if ℓ∈I\ell\in I and 00 otherwise and then define H*(X;I)=H*(X;AI)H_\ast(X;I) = H_\ast(X;A_I). If I∩𝕃X={ℓ}I \cap \mathbb{L}_X = \{\ell\}, then H*(X;I)=H*(X;ℓ)H_\ast(X;I) = H_\ast(X;\ell).

These intervalic homology groups are related by long exact sequences: if I=I1⊔I2I=I_1 \sqcup I_2 with I1<I2I_1 \lt I_2 (for instance, (a,c]=(a,b]⊔(b,c](a,c] = (a,b] \sqcup (b,c]), then we have an SES AI1→AI→AI2A_{I_1} \to A_I \to A_{I_2}, leading to an LES in homology. More generally, if I=I1⊔⋯⊔InI = I_1\sqcup \cdots \sqcup I_n with Ij<Ij+1I_j \lt I_{j+1}, then AIA_I has an nn-stage filtration, giving rise to to a spectral sequence converging from the H*(X;Ij)H_\ast(X;I_j) to H*(X;I)H_\ast(X;I). Thus, for finite metric spaces, we can regard H*(X;I)H_\ast(X;I) as built up from the H*(X;ℓ)H_\ast(X;\ell) for all ℓ∈I\ell\in I, not as a direct sum but with some cancellation coming from the connecting maps in the LES, i.e. the differentials in the spectral sequence.

Of course, this is just a fancy way of saying that there are more interesting differentials in MC⊗AIMC\otimes A_I than in the individual MC⊗AℓMC\otimes A_\ell, so that we get more cancellation, with homological algebra giving us a precise way to calculate what gets canceled. The real question is whether H*(X;I)H_\ast(X;I) is geometrically interesting, but I expect it will be.

For instance, Euler characteristics of graded groups are additive on long exact sequences. Thus, the Euler characteristic of H*(X;I)H_\ast(X;I) is the sum of all the Euler characteristics of H*(X;ℓ)H_\ast(X;\ell) for ℓ∈I\ell\in I, i.e. the numbers ck,ℓc_{k,\ell} that feed into the expected comparison with the magnitude Hahn series. Of course information is lost, since in the series the ck,ℓc_{k,\ell} for different ℓ\ells appear as coefficients of different powers of qq and so are kept separate. But qℓq^\ell is a continuous function of ℓ\ell, so it seems that if we “bin” the values of ℓ\ell into reasonably small intervals and calculate homology with coefficients in those intervals, we might get a reasonable numerical approximation to the magnitude, that might be more calculable for large finite metric spaces. More speculatively, it seems from the results you mentioned as if a lot of the geometric information about magnitude is carried by its asymptotic behavior, and that might still be extractable directly from such an approximate version.

Re: A Survey of Magnitude

I keep mentioning the Master’s thesis of Aaron Greenspan, who worked with me in 2014–15. We spent a while trying to find a magnitude homology theory for enriched categories, extending Hepworth and Willerton’s theory for graphs. We didn’t succeed, but we did make some progress, and a few of the ideas and constructions that have come up in this thread were also a part of Aaron’s work back then.

Aaron’s MSc thesis is here. I’d thought it would be publicly available via Edinburgh library, but it seems they only do that for PhD theses, not Master’s theses.

Re: A Survey of Magnitude

Thanks! I see that my proposal is an elaboration of his option (2) on pages 22-23, with semicartesianness (or more generally a two-sided bar construction) stuck in to solve the problem of missing outer face maps.

Re: A Survey of Magnitude

After an email exchange with Peter May to set myself straight on equivariant homology, I have a new perspective on this homology theory: the number ℓ\ell should not be regarded as a grading, but instead as the coefficients.

Let me start from scratch with the definition, slightly generalized. Let VV be a monoidal category, and assume for simplicity that it is small. Let CC be a small VV-category equipped with an augmentation C→IC\to I, where II is the unit VV-category with one object whose endomorphism-object is the unit I∈VI\in V. (If VV is semicartesian as before, then II is the terminal VV-category, so every VV-category is uniquely augmented.)

Define a functor MS:Vop→sSetMS:V^{op} \to sSet such that MS(ℓ)kMS(\ell)_k is the set of (k+1)(k+1)-tuples of objects x0,…,xn∈Cx_0,\dots,x_n \in C equipped with a map ℓ→C(xn−1,xn)⊗⋯⊗C(x0,x1)\ell \to C(x_{n-1},x_n) \otimes \cdots\otimes C(x_0,x_1) in VV. The degeneracy maps come from duplicating objects and using identity morphisms in CC, the inner face maps from composition in CC, and the outer face maps from the augmentation. Apply the free abelian group functor and then make simplicial abelian groups into chain complexes in either of the equivalent ways, so we get a functor MC:Vop→ChMC:V^{op}\to Ch to the category of chain complexes.

Now suppose we have a covariant functor A:V→AbA:V\to Ab (or more generally to ChCh, where AbAb is regarded as sitting in ChCh concentrated in degree 00). Then because VV is small and ChCh is cocomplete, we can form the tensor product of functors MC⊗VA∈ChMC \otimes_V A\in Ch. The homology of CC with coefficients in AA is the homology of this chain complex, H*(C;A)H_\ast(C;A). (The dual construction for an A:Vop→AbA:V^{op}\to Ab gives cohomology.)

If VV is large, then MC⊗VAMC \otimes_V A may not exist in general, but it will exist if AA is LaniA′Lan_i A' for some small subcategory i:V′→Vi:V' \to V and A′:V′→ChA' : V'\to Ch, since then MC⊗VLaniA′≅i*MC⊗V′A′MC \otimes_V Lan_i A' \cong i^* MC \otimes_{V'} A'. For example:

If V=SetV=Set and V′={1}V' = \{1\}, then A′A' is just an abelian group, and this reduces to the usual homology of (the nerve of) a category with coefficients in an abelian group.

If V=GTopV=G Top and V′V' is the orbit category of GG, then A′A' is called a coefficient system and this is the Bredon homology of (the nerve of) a GG-category.

Now let’s consider the case V=(ℕ,≥,+,0)V = (\mathbb{N},\ge,+,0) relevant for graphs. Then for any ℓ∈ℕ\ell\in\mathbb{N} we have a coefficient module Aℓ:V→AbA_\ell:V\to Ab defined by Aℓ(ℓ)=ℤA_\ell(\ell) = \mathbb{Z} and Aℓ(m)=0A_\ell(m) = 0 for m≠ℓm\neq \ell. Suppose we use the nondegenerate version of MCMC, so that MC(ℓ)MC(\ell) is Tom’s C*,≤ℓndC^{nd}_{\ast,\le\ell}. Then the tensor product MC⊗VAℓMC \otimes_V A_\ell is a quotient of ⨁mMC(m)⊗Aℓ(m)\bigoplus_m MC(m) \otimes A_\ell(m); but by definition of AℓA_\ell, all of these summands vanish except when m=ℓm=\ell, so we just have a quotient of MC(ℓ)MC(\ell).

The quotient says in general that whenever we have a morphism u:n→mu:n\to m in VV, the following two maps are to be coequalized:
MC(m)⊗Aℓ(n)→MC(n)⊗Aℓ(n)→⨁mMC(m)⊗Aℓ(m) MC(m) \otimes A_\ell(n) \to MC(n) \otimes A_\ell(n) \to \bigoplus_m MC(m) \otimes A_\ell(m) MC(m)⊗Aℓ(n)→MC(m)⊗Aℓ(m)→⨁mMC(m)⊗Aℓ(m) MC(m) \otimes A_\ell(n) \to MC(m) \otimes A_\ell(m) \to \bigoplus_m MC(m) \otimes A_\ell(m)
In our case, this can only be nontrivial if n=ℓn=\ell, since otherwise the domain of both maps is 00. The condition is also trivial if m=ℓm=\ell too, so assume m≠ℓm\neq \ell, hence (by uu) m<ℓm\lt \ell. In particular, Aℓ(m)=0A_\ell(m)=0, so the second of the above morphisms is 00; while the first one reduces to the map MC(m)→MC(ℓ)MC(m) \to MC(\ell). Thus, we are killing all the simplices of length <ℓ\lt\ell, yielding exactly Tom’s C*,=ℓndC^{nd}_{\ast,=\ell}, which is the Hepworth-Willerton magnitude chain complex. So the magnitude homology of a graph CC is precisely H*(C;Aℓ)H_\ast(C;A_\ell) as ℓ\ell varies over natural numbers, which we could write a bit abusively as H*(C;ℓ)H_\ast(C;\ell).

However, none of this depended on the numbers being integers! We can do exactly the same thing when V=[0,∞]V=[0,\infty] for metric spaces, defining a coefficient module AℓA_\ell and obtaining homology theories H*(C;ℓ)H_\ast(C;\ell) whose Euler characteristics are ∑k≥0(−1)kck,ℓ\sum_{k\ge 0} (-1)^k c_{k,\ell}. We don’t need to talk about “grading by morphisms of VV” and “approaching ℓ\ell from below”; we just choose an appropriate coefficient module. (The “graded by morphisms” version corresponds to starting with the “representable modules” FℓF_\ell defined by Fℓ(m)=V(ℓ,m)⋅ℤF_\ell(m) = V(\ell,m) \cdot \mathbb{Z} and then taking the cokernel of the map Fℓ1→Fℓ2F_{\ell_1}\to F_{\ell_2} induced by a morphism ℓ2→ℓ1\ell_2\to \ell_1 as our coefficient module; in the case of graphs the relevant AℓA_\ell can be obtained directly this way, but in the case of metric spaces it can’t.)

Re: A Survey of Magnitude

I’ve been saving up this comment to read later, as this new perspective looks extremely interesting. But for now, before I tear myself away to do some administrative stuff, here’s a small observation.

Let me start from scratch with the definition, slightly generalized. Let VV be a monoidal category, and assume for simplicity that it is small. Let CC be a small VV-category equipped with an augmentation C→IC\to I, where II is the unit VV-category with one object whose endomorphism-object is the unit I∈VI\in V. (If VV is semicartesian as before, then II is the terminal VV-category, so every VV-category is uniquely augmented.)

I’m not convinced this is genuinely more general than your earlier assumption that VV is a semicartesian monoidal category, for the following reason.

Let VV be any monoidal category. Since its unit object II is a monoid in VV, the slice category V/IV/I inherits a monoidal structure. It’s actually a semicartesian monoidal category. (It must in some sense be the universal semicartesian monoidal category on VV.) And if I’m not mistaken, a (V/I)(V/I)-category is exactly a VV-category equipped with an augmentation in your sense above.

Re: A Survey of Magnitude

The forgetful functor Σ:M/I→M\Sigma: M/I \to M is strong monoidal, and is universal in the sense of exhibiting the fact that semicartesian monoidal functors and strong monoidal functors form a coreflective sub-bicategory of the bicategory of monoidal categories and strong monoidal functors. (Check this.)

Re: A Survey of Magnitude

My first reaction to your semicartesian hypothesis was “what about Ab?” We could be trying to do the homology of rings, for instance. Your setup would make us actually look at the homology of augmented rings.

Re: A Survey of Magnitude

Re: A Survey of Magnitude

Thanks for the pointer! A quick glance on Google books tells me that for Cartan-Eilenberg, an “augmented ring” consists of a ring RR, a left RR-module QQ, and an “RR-epimorphism” (not sure exactly what that means, or whether it’s important) ε:R→Q\varepsilon : R \to Q. Then they define the homology with coefficients in a rightRR-module AA as the derived functors of A↦A⊗RQA \mapsto A\otimes_R Q.

This suggests to me that instead of an augmented VV-category CC as I originally defined it, we should be considering a VV-category CC together with two VV-functors F:Cop→VF:C^{op}\to V and G:C→VG:C\to V. The case I called “augmented” before is the one when FF and GG are constant at I∈VI\in V. Then the tensor product appearing in the definition of MSMS will be

In other words, a two-sided bar construction. My advisor Peter May used to joke that the two-sided bar construction was the only trick he knew. Sometimes I feel like I’ve inherited that from him, especially when it turns up unexpectedly like this.

What we’re seeing here is a little different from a classical two-sided bar construction, though, because there’s a Yoneda embedding happening at a funny place. Normally the two-sided bar construction would be a simplicial object whose nn-simplices are the coproduct

But here we’re first applying the Yoneda embedding to the individual tensor products, then taking the coproduct in the presheaf category SetVopSet^{V^{op}}. I suppose we could say that this is just the ordinary two-sided bar construction for SetVopSet^{V^{op}}-categories, which we get to applying the Yoneda embedding homwise to our VV-categories (which is strong monoidal when SetVopSet^{V^{op}} has the Day convolution product). Furthermore, in our case we have the additional steps of taking free abelian groups and then tensoring with “coefficients” A:V→AbA:V\to Ab, whereas in homological algebra one would generally start out in AbAb and stay there.

If we really want one definition that includes magnitude homology and also some kind of classical construction on rings from homological algebra, the best I can think of right now is to use the perhaps-excessively-general notion of bar construction from section 23 of the unpublished part of my thesis that never seems to die. If WW is a monoidal category and C∈WC\in W a monoid, a functor K:W→PK:W \to P (there’s a typo in the paper) is called a CC-bimodule if it has natural transformations K(C⊗X)→K(X)K(C\otimes X) \to K(X) and K(X⊗C)→XK(X\otimes C) \to X that are individually associative and unital, and commute with each other. Then we have a simplicial bar construction Bn(C,K)=K(C⊗n)B_n(C,K) = K(C^{\otimes n}), and if P=AbP=Ab this is a simplicial abelian group, gives rise to a chain complex, and hence a homology H*(C;K)H_\ast(C;K).

On one hand, take W=(SetVop)O×OW=(Set^{V^op})^{O\times O}, the category of O×OO\times O-matrices in SetVopSet^{V^{op}}, with the monoidal structure such that CC is a SetVopSet^{V^{op}}-category with object set OO (this is the endohom-category of OO in the bicategory of SetVopSet^{V^{op}}-matrices). Then if VV is semicartesian (or CC is augmented) and A:V→AbA:V\to Ab, defining K(X)=(∑x,yX(x,y))⊗VAK(X) = (\sum_{x,y} X(x,y)) \otimes_V A gives the homology H*(C;A)H_\ast(C;A) as in my last proposal.

On the other hand, take W=AbW=Ab, so that CC is a ring, and K(X)=A⊗X⊗QK(X) = A\otimes X \otimes Q for a left CC-module QQ and a right CC-module AA. Then I think H*(C;K)H_\ast(C;K) is the “relative Tor” Tor*C/ℤ(A,Q)Tor^{C/\mathbb{Z}}_\ast(A,Q). This probably isn’t the same as the homology Cartan-Eilenberg are defining, but at least it’s something known in homological algebra.

I’m not sure this is worth studying in this generality, but seeing it makes the semicartesian/augmentedness make a little more sense to me.

Re: A Survey of Magnitude

BUT, although an augmented VV-category is the same as a (V/I)(V/I)-category, it’s not clear that the two perspectives give the same homology. In fact, the coefficients live in different categories (functors defined on VV or on (V/I)(V/I) respectively), so we would first need to decide how to compare them before even asking the question.

Re: A Survey of Magnitude

Aha, good. I did have this general possibility in mind when I wrote my original comment about V/IV/I, which is why I put it no more strongly than “I’m not convinced”. But I didn’t revisit that thought after going through your definition of homology in detail.

This comment thread has so many interesting fronts that it’s hard to keep track of them all. I’m going to try to dispense with one of them by doing a quick post devoted to semicartesian monoidal categories.

Re: A Survey of Magnitude

Mike, this is really beautiful!

I’ve finally found time to work through it all in detail, and I think the nicest moment was seeing the Hepworth–Willerton differential emerge naturally from the general theory. I believe Richard, Simon and I have all, at times, found their formula for the differential a bit odd (though effective). However, seeing it simply pop out of your framework justifies it in a very compelling way.

Here are a few small comments.

You begin by creating, from an augmented VV-category and an object ℓ\ell of VV, a simplicial set. To do this, you have to make a distinction between inner and outer faces (using the augmentation for the latter).

This reminds me of the fact that for cartesian monoidal categories 𝒮\mathcal{S},
[Δop,𝒮]≃Colax(𝔻,𝒮)
{}[\Delta^{op}, \mathcal{S}]
\simeq
Colax(\mathbb{D}, \mathcal{S})
where 𝔻\mathbb{D} is the category of possibly empty finite ordinals n={1,…,n}\mathbf{n} = \{1, \ldots, n\} and the right-hand side is the category of colax monoidal functors from (𝔻,+,0)(\mathbb{D}, +, \mathbf{0}) to (𝒮,×,1)(\mathcal{S}, \times, 1). The easiest way to get an approximate sense of why this is true is to draw the kind of diagrams people often draw for (co)simplicial objects, with back-and-forth arrows.

The reason why I’m mentioning this is that a “simplicial object without the outer face maps” amounts to a plain ordinary functor 𝔻→𝒮\mathbb{D} \to \mathcal{S}, and then adding in the outer face maps amounts to adding the colax monoidal structure. So in this correspondence, the outer and inner face maps play different roles — as they do in your construction too.

A slightly different perspective on your construction of the simplicial set is as follows. For any VV-category AA and ℓ∈V\ell \in V we get a functor 𝔻→Set\mathbb{D} \to Set, given by
n↦∑a0,…,anV(ℓ,A(a0,a1)⊗⋯⊗A(an−1,an))
\mathbf{n} \mapsto
\sum_{a_0, \ldots, a_n} V(\ell, A(a_0, a_1) \otimes\cdots\otimes A(a_{n-1}, a_n))
When AA has an augmentation (or VV is semicartesian), you can give this functor 𝔻→Set\mathbb{D} \to Set a colax monoidal structure. In other words, you can make it into a simplicial set — and that’s exactly your MS(ℓ)MS(\ell).

I’m dwelling on this point not because there’s anything difficult or deep about the construction of this simplicial set, but in the hope that shifting perspective on what a simplicial object is might — eventually — help with the question of what to do when VV is not semicartesian.

When Richard Hepworth first started working on magnitude homology for graphs, he was for a while more interested in magnitude cohomology. That concept never made it into the published paper, and I’ve forgotten the definition, though I do have notes somewhere from those early days. Since your framework also produces a cohomology theory, it would obviously be interesting to see how the two compare.

One subtlety about magnitude homology of graphs (or metric spaces more generally) is the possibility of infinite distances. Since we’d like to be able to work with disconnected graphs, or directed graphs whose underlying poset is nontrivial, we should enrich in ℕ∪{∞}\mathbb{N} \cup \{\infty\} rather than ℕ\mathbb{N}. That doesn’t affect anything you’ve written, but Richard and Simon do stick to finite ℓ\ell in their work. A tiny lemma is needed in order to see that their definition of the differential ∂\partial (Definition 2) makes sense: that, adopting their notation, if x0≠⋯≠xkx_0 \neq \cdots \neq x_k and ℓ(x0,…,xk)=ℓ=ℓ(x0,…,xi^,…,xk)\ell(x_0, \ldots, x_k) = \ell = \ell(x_0, \ldots, \widehat{x_i}, \ldots, x_k) then xi−1≠xi+1x_{i - 1} \neq x_{i + 1}. The proof relies on ℓ\ell being finite. When ℓ=∞\ell = \infty, the definition will need tweaking a bit, and your general theory will indicate the correct thing to say.

When you obtain the magnitude homology of graphs as a special case of your approach, one of the steps is to assign to each ℓ∈ℕ\ell \in \mathbb{N} a certain functor Aℓ:ℕ→AbA_\ell \colon \mathbb{N} \to Ab. Do you have any insight into what that assignation “is” or “means”, beyond just being a simple thing one can do?

Re: A Survey of Magnitude

It reminded me of that too! But I hadn’t mentioned it before because I was failing to find the reference for that fact. I feel like it’s in one of your papers, but I can’t remember which.

It would be easier to see the way to a non-semicartesian generalization if we had some example we were trying to reproduce. For instance, is there a homology theory for rings that one might reasonably hope to reproduce in this way?

Do you have any insight into what that assignation “is” or “means”…?

The closest I’ve been able to come is that it’s a kind of “δ\delta-function at ℓ\ell”, which we’re “integrating” against MCMC to “extract its value at ℓ\ell.” But I haven’t been able to push that vague analogy into anything more precise.

Re: A Survey of Magnitude

make simplicial abelian groups into chain complexes in either of the equivalent ways, [to] get a functor MC:Vop→ChMC\colon V^{op} \to Ch.

That’s fine. There’s a normalized version of MCMC and there’s an unnormalized version of MCMC, and they’re quasi-isomorphic.

But later in the construction, we tensor MCMC with another functor A:V→ChA \colon V \to Ch and look at the homology of MC⊗VAMC \otimes_V A. You seem to take it as read that this homology is the same whichever version of MCMC we use.

I think I have a proof that this is true, using the Künneth theorem and the fact that torsion-free abelian groups are flat. But I’m guessing that my reasoning is unnecessarily elaborate. What argument did you have in mind?

Re: A Survey of Magnitude

Hmm, to be honest I didn’t think about that point. We definitely want the operation MC⊗VAMC\otimes_V A to be homotopy invariant, not just so that it doesn’t depend on which version of MCMC we use, but I expect we’d also need that to show that magnitude homology is invariant under equivalence of enriched categories. If it isn’t homotopy-invariant, then we should use a derived version instead. (With our fully ∞\infty-categorical hats on, we should be bypassing chain complexes entirely, regarding AA as a diagram of spectra and smashing it with MSMS and then taking homotopy groups.)

But of course we want to have concrete models to do calculations with. Both MCMC’s consist of free abelian groups, which are therefore projective and flat, but that doesn’t necessarily make them cofibrant as VopV^{op}-diagrams; so I’m not immediately sure how to go about it. I’d like to see your proof.

Re: A Survey of Magnitude

The argument I had in mind when I wrote my last comment wasn’t quite
complete, but I think I’ve got it now.

Write MCMC for the unnormalized version and MCndMC^{nd} for the normalized
version. These are both functors Vop→ChV^{op} \to Ch. I’ll show that the
chain complexes MC⊗VAMC \otimes_V A and MCnd⊗VAMC^{nd} \otimes_V A are
quasi-isomorphic for any functor A:V→ChA \colon V \to Ch.

In this argument, all I’ll use about MCMC and MCndMC^{nd} are the following two properties:

MCMC and MCndMC^{nd} are obtained from some functor from VopV^{op} to (simplicial abelian groups) by
applying the unnormalized and normalized chain complex constructions,
respectively.

For each ℓ∈V\ell \in V and n∈ℕn \in \mathbb{N}, the abelian group
MC(ℓ)nMC(\ell)_n is free. (Or if you want to be more parsimonious,
torsion-free.)

Here goes.

For ℓ∈V\ell \in V, let D(ℓ)D(\ell) be the subcomplex of MC(ℓ)MC(\ell)
generated by the degenerate elements. General results about simplicial
abelian groups tell us that MC(ℓ)=MCnd(ℓ)⊕D(ℓ)MC(\ell) = MC^{nd}(\ell) \oplus D(\ell) and
that D(ℓ)D(\ell) is acyclic. Moreover, DD is a subfunctor of MCMC, so MC=MCnd⊕DMC =
MC^{nd} \oplus D in the functor category [Vop,Ch][V^{op}, Ch].

Claim: for each ℓ∈V\ell \in V and n∈ℕn \in \mathbb{N}, every subgroup of
the abelian group D(ℓ)nD(\ell)_n is flat. Indeed, the group MC(ℓ)nMC(\ell)_n is
free and in particular torsion-free, and D(ℓ)nD(\ell)_n is a subgroup of it,
so every subgroup of D(ℓ)nD(\ell)_n is torsion-free. Then use the fact that
for abelian groups (or more generally modules over a PID), flatness is
equivalent to torsion-freeness (Exercise 3.2.3 of Weibel’s book, for
instance).

Next we’ll show that D(ℓ)⊗QD(\ell) \otimes Q is acyclic for each ℓ∈V\ell \in
V and chain complex QQ. One version of Künneth’s theorem (Weibel’s
Theorem 3.6.3) tells us that for any chain complexes PP and QQ over a
commutative ring, and any nn, the sequence
0→⨁p+q=nHp(P)⊗Hq(Q)→Hn(P⊗Q)→⨁p+q=n−1Tor1(Hp(P),Hq(Q)→0
0 \to
\bigoplus_{p + q = n} H_p(P) \otimes H_q(Q) \to
H_n(P \otimes Q) \to
\bigoplus_{p + q = n - 1} Tor_1(H_p(P), H_q(Q) \to
0
is exact, as long asPnP_n and dPnd P_n are flat for each nn. It
follows that if PP satisfies these flatness hypotheses and is acyclic
then P⊗QP \otimes Q is also acyclic. By (2), P=D(ℓ)P = D(\ell) does satisfy the
flatness hypotheses, and it’s acyclic by (1), so D(ℓ)⊗QD(\ell) \otimes Q is acyclic.

In the category of chain complexes, the class of acyclic complexes is
closed under small colimits. This must be well-known, but I didn’t know
it, so here’s the proof I came up with. It suffices to show that it’s
closed under coproducts and cokernels. Coproducts are easy. For
cokernels, take a short exact sequence of complexes whose first two terms
are acyclic. Then in the homology long exact sequence, two out of every
three consecutive terms are zero, from which it follows that they’re all
zero.

Now let A:V→ChA\colon V \to Ch be any functor. I claim that the chain
complex
D⊗VA=∫ℓ∈VD(ℓ)⊗A(ℓ)
D \otimes_V A = \int^{\ell \in V} D(\ell) \otimes A(\ell)
is acyclic. Like any other coend, this can be expressed as a colimit, and
in the diagram that we’re taking the colimit of, all the objects are of the
form D(ℓ)⊗A(m)D(\ell) \otimes A(m) for some ℓ,m∈V\ell, m \in V. By (3), all these
objects are acyclic, so by (4), their colimit D⊗VAD \otimes_V A is acyclic.

By (1),
MC⊗VA=(MCnd⊗VA)⊕(D⊗VA)
MC \otimes_V A
=
\Bigl( MC^{nd} \otimes_V A \Bigr)
\oplus
\Bigl( D \otimes_V A \Bigr)
in ChCh. Apply the functor H*H_\ast to both sides: then by (5), the maps H*(MC⊗VA)⇆H*(MCnd⊗VA)H_\ast(MC
\otimes_V A) \leftrightarrows H_\ast(MC^{nd} \otimes_V A) induced by the
projection and inclusion MC⇆MCndMC \leftrightarrows MC^{nd} are mutually
inverse. In particular, MC⊗VAMC \otimes_V A and MCnd⊗VAMC^{nd} \otimes_V A are
quasi-isomorphic.

I guess some simplification is possible in the case where A:V→ChA\colon V \to Ch is concentrated in degree zero.

Re: A Survey of Magnitude

Are you sure about (4)? It’s weird if true. The class of contractible spaces, for instance, is very far from closed under colimits — its colimit completion includes all CW complexes.

In your proof, the homology LES usually comes from an SES of chain complexes, which means that the map you’re taking the cokernel of must be mono. So don’t you need an extra fact like the image of a map between acyclic complexes is acyclic?

Re: A Survey of Magnitude

OK, (4) is definitely false, because the cokernel of a map between acyclic complexes can fail to be acyclic. For example, let AA be any nontrivial abelian group, let CC be the complex
⋯→0→0→A→A→0→⋯
\cdots \to 0 \to 0 \to A \to A \to 0 \to \cdots
with the copies of AA in degrees 00 and 11, and let DD be the complex
⋯→0→A→A→0→0→⋯
\cdots \to 0 \to A \to A \to 0 \to 0 \to \cdots
with the copies of AA in degrees −1-1 and 00. The unlabelled maps are identities. Take the map C→DC \to D that’s the identity in degree 00. Its cokernel EE is
⋯→0→A→0→0→0→⋯.
\cdots \to 0 \to A \to 0 \to 0 \to 0 \to \cdots.
Thus, CC and DD are acyclic, but EE isn’t.

Re: A Survey of Magnitude

Yes, what is true is that acyclic complexes are closed under taking mapping cones, i.e. under homotopy co-kernels. This illustrates Mike’s point: one could only expect the on-the-nose co-kernel to be quasi-isomorphic to the mapping cone if the morphism of acyclic complexes one begins with is a cofibration, which in particular entails that it is a monomorphism.

Re: A Survey of Magnitude

So, we still don’t have a proof that the magnitude homology is independent of which version of MCMC you use.

I’m beginning to doubt it. The topological analogy that Mike invokes suggests that in my step (5), the chain complex D⊗VAD \otimes_V A needn’t be acyclic. For comparison: if D:Δ→TopD \colon \Delta \to Top is the cosimplicial space consisting of the standard topological simplices, and AA is any simplicial space, then A⊗ΔDA \otimes_\Delta D is the geometric realization of AA, which usually isn’t contractible even though every D(n)D(n) is.

If the chain complex D⊗VAD \otimes_V A isn’t acyclic then MCnd⊗VAMC^{nd} \otimes_V A and MC⊗VAMC \otimes_V A have different homologies, since
H*(MC⊗VA)=H*(D⊗VA)⊕H*(MCnd⊗VA).
H_\ast(MC \otimes_V A) = H_\ast(D \otimes_V A) \oplus H_\ast(MC^{nd} \otimes_V A).
In other words, the magnitude homology does depend on which version of MCMC you use.

Re: A Survey of Magnitude

The point that I made in my previous comment suggests a remedy: replace the tensor product by a ‘homotopy tensor product’ (i.e. replace the co-end with a homotopy co-end). This is not as bad as it sounds for chain complexes: it is just a double mapping cylinder-like construction using the usual ‘interval’ in chain-complexes, and can probably be described quite explicitly.

This seems more conceptually correct in any case: if one is constructing homology by means of this general construction, it should be invariant under ‘quasi-isomorphism’ of the functors involved, and for that to hold in general, one should certainly be working homotopy colimits.

Re: A Survey of Magnitude

OK, so now that we’re getting into derived territory, let me ask the question that I’ve been saving up/repressing for almost this whole conversation: does this magnitude homology theory fit into the general framework for homology described here? (Well, that’s a framework for cohomology… but dualize!)

It’s hard to see how it could, but I’m not very experienced at these things. For a start, how could a VV-category CC give rise to a topos? Thinking out loud, we have the nerve-like functor MS(C):Vop→sSetMS(C)\colon V^{op} \to sSet, so we get the slice topos [Vop,sSet]/MS(C)[V^{op}, sSet]/MS(C). But this is itself just a presheaf topos (on the category of elements of MS(C):(V×Δ)op→SetMS(C)\colon (V \times \Delta)^{op} \to \Set), so it seems unlikely that it’s going to take us in the right direction.

Re: A Survey of Magnitude

I also said in a previous comment:

If it isn’t homotopy-invariant, then we should use a derived version instead. (With our fully ∞\infty-categorical hats on, we should be bypassing chain complexes entirely, regarding AA as a diagram of spectra and smashing it with MSMS and then taking homotopy groups.)

The general framework for (co)homology described in your comment is, in my opinion, a shadow of the ∞\infty-categorical picture that people worked with before we understood the latter. It’s sometimes applicable, but I find the higher-categorical version to be more conceptual and also more general.

As David C. said, for cohomology the basic framework is very simple and very general: cohomology is just the homotopy groups of a mapping space in an (∞,1)(\infty,1)-category (usually an ∞\infty-topos), where the target (the “coefficients”) is usually a spectrum object so that you get HnH^n for all nn. A somewhat more general picture, which also dualizes better, is to have a 2-variable (∞,1)(\infty,1)-adjunction (⊗,homl,homr):∞Gpd×C→D(\otimes,\hom_l,\hom_r):\infty Gpd \times C \to D; then the cohomology of X∈CX\in C with coefficients in A∈DA\in D is the homotopy groups of homr(X,A)\hom_r(X,A). The dual picture is then a 2-variable (∞,1)(\infty,1)-adjunction (⊗,homl,homr):C×D→∞Gpd(\otimes,\hom_l,\hom_r):C \times D \to\infty Gpd, so that the homology of X∈CX\in C with coefficients in A∈DA\in D is the homotopy groups of X⊗AX\otimes A.

To be more explicit in our case, I would regard MS:Vop→sSetMS:V^{op} \to sSet as an object of the ∞\infty-topos ∞GpdVop\infty Gpd^{V^{op}}, or perhaps (worrying about size) ∞Gpd(V′)op\infty Gpd^{(V')^{op}} for some small subcategory V′⊂VV'\subset V. Then if AA is a spectrum object in this ∞\infty-topos, the hom-space MapV(MS,A)Map_V(MS,A) is a spectrum, and its homotopy groups are the cohomology H*(MS;A)H^\ast(MS;A). Similarly, if AA is a spectrum object in the dual ∞\infty-topos ∞GpdV\infty Gpd^{V}, the smash product MS∧VAMS \wedge_V A is a spectrum, and its homotopy groups are the homology H*(MS;A)H_\ast(MS;A).

Re: A Survey of Magnitude

However, with all of that said, the explicit connection to the magnitude function comes from the specific chain complex MCnd⊗VAMC^{nd}\otimes_V A. So even if the general definition is done in a directly invariant way, at some point we need to come back to these explicit constructions (or similar ones) in order to show that the desideratum for “magnitude homology” is satisfied.

For whatever it’s worth, note that the analogy with contractible spaces isn’t exact. In particular, the chain complex associated to a contractible space isn’t acyclic; it has H0=ℤH_0 = \mathbb{Z}. The chain complex of a space is only acyclic if the space is empty, and empty spaces are closed under colimits. Of course, as your example shows, acyclic complexes aren’t closed under colimits…

I bet that equivariant homotopy theorists must know the answer to this question in some sense, since Bredon homology and cohomology are a special case of this definition. But we might have trouble communicating well enough to get the answer out of them.

Re: A Survey of Magnitude

Just a few remarks. Firstly, I don’t think that there is any problem with showing that the two functors in question give the same homology theory when we have an abelian group; it is only when passing to more general chain complexes that we have to work with homotopy colimits.

Secondly, I think that Tom’s question makes perfect sense as it stands when considering abelian groups, without passing to (∞,1)(\infty, 1)-categories and spectra. That is to say, I think it is very natural that there should be an interpretation of a cohomology theory where we have abelian group coefficients in the framework that Tom describes.

Thirdly, I’d like to offer one suggestion for a general way to construct a cohomology theory for VV-categories. It’s a bit off the cuff, but maybe it will suggest some ideas.

Let’s first discuss constructing a nerve functor. As long as we have a free VV-category on a directed graph (or on a category) functor, one could proceed as follows. There is a functor Δ≤2→V−Cat\Delta_{\leq 2} \rightarrow V-Cat which is the same as the usual one into CatCat used for constructing the nerve, except that for the 00-simplex and 11-simplex we apply the free VV-category functor as well, and for the 22-simplex we should I think first take the free VV-category on a triangle, and then ‘force the triangle to commute in a VV-categorical sense’, by which I mean take a colimit in V−CatV-Cat which forces the two ways around the triangle to be equal.

This functor, by the usual formalism, determines a nerve adjunction between 2-truncated simplicial sets and VV-Cat. One then combine this with the 2-coskeleton adjunction to get a full nerve adjunction. Or, if one prefers, one can do something similar to what I outlined for the 22-simplex for the nn-simplex for every nn.

This nerve functor provides us with one notion of cohomology. Can we also interpret it in a topos-theoretic way? My guess would be that we can proceed as follows. We take the VV-category of presheaves of VV-objects on our category (i.e. VV-functors from the opposite of our category to VV). We then look at what we might call VV-abelian group objects in this ‘enriched topos’ (the notion of abelian group makes perfect sense internally to a monoidal category). I expect that we get something like a VV-enriched abelian category. And probably one can carry out some kind of VV-enriched version of homological algebra to cook up a cohomology theory.

Re: A Survey of Magnitude

Firstly, I don’t think that there is any problem with showing that the two functors in question give the same homology theory when we have an abelian group

That’s great! Can you give a proof? It’s not obvious to me how to proceed, because neither the coefficient system A:V→Ab⊆ChA:V\to Ab \subseteq Ch nor the chain complex MC:Vop→ChMC : V^{op}\to Ch is obviously cofibrant in any model structure on diagrams of chain complexes.

I think it is very natural that there should be an interpretation of a cohomology theory where we have abelian group coefficients in the framework that Tom describes.

Many cohomology theories have such an interpretation, but ultimately I think it is limiting to expect that all of them will. So yes, the question makes perfect sense, but we shouldn’t be disturbed if the answer is no.

for the 0-simplex and 1-simplex we apply the free V-category functor as well, and for the 2-simplex we should I think first take the free V-category on a triangle, and then ‘force the triangle to commute in a V-categorical sense’… This functor, by the usual formalism, determines a nerve adjunction between 2-truncated simplicial sets and V-Cat.

I’m having trouble seeing how this is going to produce anything different than the nerve of the underlying ordinary category of a V-category. The free V-category on a 1-simplex, for instance, has two objects xx and yy, with hom(x,x)=hom(y,y)=hom(x,y)=I\hom(x,x)=\hom(y,y)=\hom(x,y)=I and hom(y,x)=∅\hom(y,x)=\emptyset, so a map out of it into CC is just a selection of two objects a,b∈Ca,b\in C and a morphism I→C(a,b)I \to C(a,b) in VV, i.e. a morphism from aa to bb in the underlying ordinary category of CC.

Re: A Survey of Magnitude

When I started to think about it more carefully, it turned out to be rather harder than I first thought to show that we are OK with coefficients landing in abelian groups! In general, there is certainly no hope: even in the case that our coefficients functor AA is just the constant functor on the zero group, i.e. the unit for the monoidal structure on abelian groups, the co-end will give us the colimit of the other functor, and there is no reason that a quasi-isomorphism should be preserved under colimits.

However, in the special case that you and Tom have been discussing, I think we’re OK, for arbitrary coefficient systems in fact. Here is the argument I have in mind.

We have three ways to convert a simplicial abelian group to a chain complex: the normalised chains complex; the alternating face map complex; and the latter modulo degeneracies. The alternating face map complex is chain homotopic to the normalised chains complex; and is quasi-isomorphic to the alternating face map complex modulo degeneracies. But the alternating face map complex, in the situation you are considering, is projective in every degree. This means that we can in fact replace the latter quasi-isomorphism by a chain homotopy.

Now, one can view chain homotopy as tensoring with a certain chain complex II (an interval object). In the co-end defining the tensor product of functors, we can additionally tensor C(v)⊗A(v′)C(v) \otimes A(v'), where CC is the functor coming from the alternating face map complexes, with II; and then use the individual chain homotopies for each vv to construct a morphism of chain complexes out of the co-end. Because the tensor product of chain complexes preserves colimits, the tensoring with II can be taken outside of the co-end, and so we obtain a chain homotopy between C⊗VAC \otimes_{V} A, N⊗VAN \otimes_{V} A, and D⊗VAD \otimes_{V} A, where NN is the functor coming from the normalised chains complexes, and DD is the functor coming from the alternating faces modulo degeneracies complexes.

D(A)D(A) is defined to be the chain subcomplex of C(A)C(A) generated by the images of the degeneracies σi\sigma_i, so that
Dn(A)=∑σi(Cn−1A).
D_n(A) = \sum \sigma_i(C_{n-1} A).
(I know, you used DD for something different, but I’m sticking to the book’s notation.)

Lemma 8.3.7 states that C(A)=N(A)⊕D(A)C(A) = N(A) \oplus D(A), and so N(A)N(A) is isomorphic to C(A)/D(A)C(A)/D(A).

I assume C(A)C(A) is what you’re calling the “alternating face map complex”, that N(A)N(A) is what you’re calling the “normalised chains complex”, and that C(A)/D(A)C(A)/D(A) is what you’re calling the “alternating face map complex modulo degeneracies”. If those assumptions are all correct, then the normalised chain complex is isomorphic to the alternating face map complex modulo degeneracies.

But I don’t think my assumptions about your terminology can all be correct, or else I’m misunderstanding, because otherwise it would be bizarre for you to write that there are three chain complexes in play.

Re: A Survey of Magnitude

Sorry for the confusion! I was just referring to the chain complexes defined on the nLab here. You’re entirely correct about there being an isomorphism between two of the complexes. I was taking my cue from what are 3.3 and 3.4 on the nLab page, and not reading/remembering the remainder! The only thing that is important for my argument is to be able to work with chain homotopy equivalences rather than quasi-isomorphisms.

Re: A Survey of Magnitude

Re: A Survey of Magnitude

the zero group, i.e. the unit for the monoidal structure on abelian groups

Well, the unit for the monoidal structure on abelian groups is ℤ\mathbb{Z}, not the zero group. But yes, if we take AA to be constant at ℤ\mathbb{Z}, then we get colimits.

the alternating face map complex, in the situation you are considering, is projective in every degree. This means that we can in fact replace the latter quasi-isomorphism by a chain homotopy.

True. But what we need in order for the rest of your argument to go through is that the resulting chain homotopy varies naturally in vv, i.e. it is a chain homotopy in ChVopCh^{V^{op}}. (I think the rest of the argument then essentially boils down to saying that −⊗VA-\otimes_V A is a ChCh-enriched functor, hence preserves chain homotopies.) I don’t think we get such a natural chain homotopy from a quasi-isomorphism and a levelwise projective chain complex.

However, according to the nLab page at least, the inclusion of the normalized chains into the unnormalized ones is already a natural chain homotopy equivalence. If that’s true, then maybe we are fine.

Re: A Survey of Magnitude

Well, the unit for the monoidal structure on abelian groups is ℤ\mathbb{Z} not
the zero group.

Sorry for the ‘typo’! I have very limited time for thinking about mathematics, and am usually rather tired before I even begin, which tends to lead to these kinds of ‘typos’ slipping in every now and then.

what we need in order for the rest of your argument to go through is that the
resulting chain homotopy varies naturally in vv

That’s correct, thanks! Happily, as you say, it seems that we do have that naturality.

Re: A Survey of Magnitude

Many cohomology theories have such an interpretation, but ultimately I think it is
limiting to expect that all of them will. So yes, the question makes perfect sense,
but we shouldn’t be disturbed if the answer is no.

What I was suggesting was that in the case of abelian group coefficients, we should expect the answer to be yes.

I’m having trouble seeing how this is going to produce anything different than the
nerve of the underlying ordinary category of a V-category.

What happens with the 2-simplex was the intended key to the construction. I don’t think it’s unreasonable that if we look at the 1-truncation, then we just recovers the underlying ordinary category. Can one recover any free VV-category as a colimit in VV-Cat of the free VV-categories on a 0-simplex and 1-simplex? I would expect the answer to this to be: yes.

The idea was that the 2-simplex should be sent to something which allows us to obtain non-free VV-categories by taking colimits in VV-Cat. I was attempting to define the analogue of a ‘free-standing commutative triangle’ appropriate for VV-categories.

It may be simpler to think cubically. In this case, we know what the 2-cube must be sent to: the tensor product of the free V-category on an interval with itself, which would be the analogue of a ‘free-standing commutative square’ that I am looking for in the setting of VV-categories.

So it’s nothing too deep. But it seems to me to be the right sort of picture.

Re: A Survey of Magnitude

What I was suggesting was that in the case of abelian group coefficients, we should expect the answer to be yes.

And I was saying that I disagree; I don’t see any reason to expect that. Note that for the present cohomology theory, the coefficients are never just “abelian groups”; they are diagrams of abelian groups.

Can one recover any free VV-category as a colimit in VV-Cat of the free VV-categories on a 0-simplex and 1-simplex?

If by “free VV-category” you mean the free VV-category generated by an unenriched graph, then yes, certainly. But mapping out of such free VV-categories is not going to tell you about anything other than the underlying ordinary categories of VV-categories.

I was attempting to define the analogue of a ‘free-standing commutative triangle’ appropriate for VV-categories.

The only way I can think of to interpret this is that there are three objects x,y,zx,y,z and we have C(x,x)=C(y,y)=C(z,z)=C(x,y)=C(y,z)=C(x,z)=IC(x,x)=C(y,y)=C(z,z)=C(x,y)=C(y,z)=C(x,z)=I and all other hom-objects empty. This is the free VV-category on the directed graph (⋅→⋅→⋅)(\cdot \to\cdot\to\cdot), and mapping out of it detects commutative triangles in the underlying ordinary category. Colimits involving this 2-simplex will produce VV-categories that are free on ordinary categories, and hence mapping out of them will only detect the underlying ordinary category of a VV-category.

If that’s not what you intended for the 2-simplex, then what did you intend?

Re: A Survey of Magnitude

Note that for the present cohomology theory, the coefficients are never just
“abelian groups”; they are diagrams of abelian groups.

Ah, we have been talking about different things. My attempts to define a nerve functor, and to consider Tom’s question, have been in the setting of: how to define a cohomology theory, with abelian group coefficients (where abelian group is possibly to be taken in an enriched sense), for an enriched category? There is a bit of a gap between this and the setting in which your discussion with Tom is taking place.

Regarding the nerve functor, here is an attempt to make things clearer. It seems to me that we can cook up a nerve functor from an enriched category in a very simple way. An nn-simplex consists of an (n+1)(n+1)-tuple of objects (x0,…,xn)(x_{0}, \ldots, x_{n}), and an arrow 1→Hom(x0,x1)⊗Hom(x1,x2)⋯Hom(xn−1,xn)1 \rightarrow Hom(x_{0},x_{1}) \otimes Hom(x_{1}, x_{2}) \cdots Hom(x_{n-1}, x_{n}) in VV, where 11 is the unit of the monoidal structure. The face maps are given by composition in the enriched category. The degeneracies come from the identities in the enriched category.

So this is an obvious generalisation of the usual nerve, which we recover in the case that VV is SetSet. It is also obviously closely related to your ‘parametrised nerve’ construction for homology.

We can recover this nerve functor from the kind of formalism that I have been attempting to describe. My use of a free VV-category functor, though I think it makes sense where we have it, was not necessary. In the case of the 1-simplex and 2-simplex, we get the VV-categories you describe (where I is the unit object of the monoidal structure).

One aspect of this nerve functor is that I think one can see straightforwardly that a VV-natural transformation between VV-functors gives rise to a simplicial homotopy: a VV-natural transformation should be able to be viewed as a VV functor out of the source category for the functors tensored (in the usual sense for VV-categories) with an ‘interval’ VV-category, which should be the gadget I described above to which the 1-simplex is sent. It is then I think easy to see that the nerve of this VV-functor gives rise to a simplicial homotopy.

I haven’t yet worked fully through your latest comment in which you argue that the same property holds for homology, but I hope that something like the same argument could be given in that setting, if one understands the ‘parametrised nerve’ construction in a sufficiently nice category theoretic way (which you seem to be outlining in that comment).

Re: A Survey of Magnitude

The face maps are given by composition in the enriched category.

That works for the inner face maps, but not the outer ones. This is precisely why I assumed semicartesianness of VV (or augmentedness of our VV-category), to get the outer face maps as well. When you add that, then what you describe is exactly the case ℓ=1\ell=1 of my “parametrized nerve” construction (a case that carries all the information when V=SetV=Set, but none of it when V=[0,∞]V=[0,\infty]). I don’t think it can be described using maps out of free VV-categories unless ⊗\otimes is cartesian, since a VV-functor doesn’t involve any maps into tensor products as basic data.

You may be right that homotopy-invariance is easier if we describe VV-natural transformations using tensors with the interval VV-category; I’ll need to think about that.

Re: A Survey of Magnitude

I think there are possibly two different kinds of motivating example here. Let me try to illustrate where my intuition is coming from by looking at a very concrete example, that of categories enriched over abelian groups. Everything that I will do I think works completely generally, but it may help to see it in this one case.

There is a perfectly good interval object in AbAb-Cat. It has two objects, which I’ll denote by 00 and 11; Hom(0,0)Hom(0,0), Hom(1,1)Hom(1,1), and Hom(1,0)Hom(1,0) are the zero abelian group; and Hom(0,1)Hom(0,1) is ℤ\mathbb{Z}. There is only one way to define composition and identities.

There is also a nice ‘free-standing commutative triangle’. It has three objects, which I’ll denote by 00, 11, and 22. We have that Hom(i,i)=0Hom(i,i) = 0 for all ii; that Hom(0,1)=Hom(1,2)=ℤHom(0,1) = Hom(1,2) = \mathbb{Z}; that Hom(0,2)=ℤ⊗ℤHom(0,2) = \mathbb{Z} \otimes \mathbb{Z} (of course this is isomorphic to ℤ\mathbb{Z}, but it may be more suggestive to write it like this); and that the other three Hom abelian groups are 00. Composition Hom(0,1)⊗Hom(1,2)→Hom(0,2)Hom(0,1) \otimes Hom(1,2) \rightarrow Hom(0,2) is the identity; and there is only one way to define the other composition arrows, and the identity arrows.

Now, these two gadgets assemble into a functor Δ→Ab−Cat\Delta \rightarrow Ab-Cat, sending Δ0\Delta^{0} to the ‘free-standing point’, namely the AbAb-category with a single object and with the zero group as its arrows. So we obtain a nerve adjunction to 2-truncated simplicial sets (to which we can apply the 2-coskeleton adjunction to get one to all of simplicial sets; or we can do something similar to what I have done above for the nn-simplex for any nn).

This nerve functor from Ab−CatAb-Cat to simplicial sets has exactly the explicit description that I gave in my previous comment, but AbAb is not semi-cartesian. In fact, I am making key use of the fact that AbAb is semi-cartesian, in distinguishing between ℤ\mathbb{Z} and 00.

In what generality can we carry out this construction? Always I think, if VV has an initial object: for then the role of 00 is played by this initial object; and the role of ℤ\mathbb{Z} is played by the unit of the monoidal structure. If there is no initial object, we can always just chuck one in (I’m sure this universal construction exists for monoidal categories). In other words, it seems to me that we can essentially always carry it out.

Does this nerve functor, and the corresponding cohomology theory, detect useful information? I haven’t tried to look at an example. But I don’t see why not, in some cases at least. In the case of AbAb-Cat, for instance, the nerve seems to encode all the useful information. The simple argument that I outlined for showing that a VV-natural transformation between VV-functors gives rise to a simplicial homotopy certainly goes through.

How does it compare to the ‘parametrised nerve’ construction that you outlined? I sort of see a way to motivate the latter construction: in passing from the trivial category to a non-trivial one, we might expect to work with ‘generalised elements’, so should expect a functor from VV to sSetsSet rather than one from 11 to sSetsSet, i.e. a single simplicial set. But I’m not convinced either by the general applicability of the construction, yet at least. In particular, I am not convinced by the way that it handles the case of SetSet. You explain how to recover this case, but that explanation does not seem to me to work very generally. On the other hand, of course, your construction does cover at least two interesting cases: Bredon homology, and the kind of homology theories that you have been discussing with Tom.

Re: A Survey of Magnitude

Hom(0,0)Hom(0,0), Hom(1,1)Hom(1,1), and Hom(1,0)Hom(1,0) are the zero abelian group; and Hom(0,1)Hom(0,1) is ℤ\mathbb{Z}

I think you must have meant that Hom(0,0)=Hom(1,1)=ℤHom(0,0)=Hom(1,1)=\mathbb{Z} as well. If Hom(x,x)=0Hom(x,x)=0 in an Ab-category, then xx is a zero object, which is not what you want (in particular, any morphism to or from a zero object is zero, so you couldn’t have Hom(0,1)=ℤHom(0,1)=\mathbb{Z} that way). Similarly in all your other simplices, all endomorphism objects should be ℤ\mathbb{Z}.

This nerve functor from Ab−CatAb-Cat to simplicial sets has exactly the explicit description that I gave in my previous comment,

a function on objects, selecting three objects x0,x1,x2∈Cx_0,x_1,x_2\in C

morphisms HomΔ2(i,j)→HomC(i,j)Hom_{\Delta^2}(i,j) \to Hom_C(i,j) in AbAb for all i,ji,j

such that composition and identities are preserved.

If we make the change I suggested above, then the morphisms HomΔ2(i,j)→HomC(i,j)Hom_{\Delta^2}(i,j) \to Hom_C(i,j) when i=ji=j are uniquely determined by preservation of identities. Thus, all that remains is three morphisms ℤ→C(x0,x1)\mathbb{Z}\to C(x_0,x_1) and ℤ→C(x1,x2)\mathbb{Z}\to C(x_1,x_2) and ℤ→C(x0,x2)\mathbb{Z}\to C(x_0,x_2). Preservation of composition then requires that the composite

is equal to the given morphism ℤ→C(x0,x2)\mathbb{Z}\to C(x_0,x_2). Thus, the data is exactly just ℤ→C(x0,x1)\mathbb{Z}\to C(x_0,x_1) and ℤ→C(x1,x2)\mathbb{Z}\to C(x_1,x_2), i.e. a pair of composable morphisms in the underlying ordinary category of CC. So, as I said, you are really just recovering the ordinary nerve of the underlying ordinary category, not noticing anything about the enrichment.

In particular, I am not convinced by the way that it handles the case of Set. You explain how to recover this case, but that explanation does not seem to me to work very generally.

I’m not sure what sort of additional generality you’re looking for.

In fact, I am making key use of the fact that Ab is not semi-cartesian, in distinguishing between ℤ\mathbb{Z} and 0

Actually, I think what you’re using there is the fact that Ab is not semi-cocartesian. Even in a semicartesian category the unit object may be (and usually is) different from the initial object, and as you said the main thing you’re using is that there is an initial object (which I think you also need to have preserved by the tensor product on both sides).

Re: A Survey of Magnitude

I think you must have meant

I was actually just not thinking carefully enough, but thanks for putting it delicately! In a sense, it doesn’t really matter, because we are just throwing the identites in; but the issue with the 2-simplex is more serious, thanks for remarking upon it.

Though my efforts have so far gone up in smoke, I still think that there should be a way to think about things in the way I am trying to do. Let me try once again :-).

Something that I have been thinking all along, but for some reason had been trying to avoid, is that the nerve of an VV-enriched category should land in VΔopV^{\Delta^{op}}, that is, simplicial objects in VV. To be more precise, I think we could replace Δ\Delta by the free VV-category F(Δ)F(\Delta) on Δ\Delta. Then, under sufficiently nice conditions on VV (complete, co-complete, closed symmetrical monoidal), we would have that the enriched functor category from F(Δ)opF(\Delta)^{op} to VV would be the free co-completion of F(Δ)F(\Delta). Then one should have a VV-enriched analogue of the usual formalism that allows one to construct a nerve adjunction from a functor from Δ\Delta to some co-complete category; this would use the internal Hom in VV. By replacing a small VV by the category of presheaves of sets upon it with the Day convolution monoidal structure, we should not lose any generality compared to the examples that you have been considering.

If one uses this kind of formalism, my guess would be that the naïve nerve functor formalism that I have been trying to get to work would in fact now work. Is that incorrect?

If it works, one would now use abelian groups and chain complexes internal to VV to complete the construction of a cohomology theory (as cropped up in one of my first comments in this thread).

I would like to raise one another construction for discussion. If I am not mistaken, one can define a ‘Hochschild cohomology’-like simplicial set for any category enriched over a symmetric monoidal category. An nn-simplex consists of a set of n+1n+1 objects x0,…,xnx_{0}, \ldots, x_{n} together with an arrow 1→Hom(x0,x1)⊗⋯⊗Hom(xn−1,xn)⊗Hom(xn,x0)1 \rightarrow Hom(x_{0}, x_{1}) \otimes \cdots \otimes Hom(x_{n-1},x_{n}) \otimes Hom(x_{n}, x_{0}). The faces are given by composition between consecutive pairs in this tensor product, except for the last one, in which we rearrange the tensor product to look like Hom(xn,x0)⊗Hom(x0,x1)⊗⋯⊗Hom(xn−1,xn)Hom(x_{n}, x_{0}) \otimes Hom(x_{0}, x_{1}) \otimes \cdots \otimes Hom(x_{n-1},x_{n}) and then compose the first pair in the tensor product.

When VV is AbAb, and our category has only one object, we recover the underlying simplicial set of the simplicial abelian group giving rise to Hochschild cohomology. I don’t know whether the cohomology of this simplicial set itself is interesting. If it isn’t, we could replace simplicial sets by simplicial VV-objects in the above construction, in a similar manner as earlier in this comment, so that we will exactly recover Hochschild cohomology in the AbAb-enriched one object case; but I’m not sure in general whether this is useful, because I’m not sure that we will be able to pass from a simplicial VV-object to a chain complex in a useful way.

Re: A Survey of Magnitude

Re: A Survey of Magnitude

I am already making quite strong assumptions about VV (but they are weak enough to admit many interesting examples, including enrichment over any small symmetric monoidal category, due to Day convolution) to ensure that we have a free co-completion in the VV-enriched sense. In particular, I am already assuming that VVis complete, so we do have that V−CatV-Cat is enriched over itself.

Re: A Survey of Magnitude

Maybe everyone’s tired of the discussion by now, but since you didn’t remark upon it, let me draw attention again to the Hochschild cohomology-like construction in its own comment. One can play exactly the same ‘Bredon homology-like’ game with this construction as you do with your construction: one can cook up a functor Vop→sSetV^{op} \rightarrow sSet in exactly the same way, one can tensor it with a coefficient functor, etc. And if you choose the same coefficient system as you do to get magnitude homology, exactly the same calculation as you made will give us something that is very similar to magnitude homology, except that our points x0x_{0}, …\ldots, xnx_{n} are now arranged in a ‘circle’. So we will have something like a Hochschild homology/cohomology, with a circle action, etc, for graphs, and so on.

Moreover, this construction works completely generally; there is no requirement of semi-cartesianness.

Filtered spaces: Persistent homology is about the effect of applying the homology functor to a filtration of topological spaces. Invariants describing the resulting homology diagrams help us construct tools for visualization and data analysis eventually allowing for the inference of topological structure
for point clouds using specific constructions of filtered complexes that encode properties of point clouds.

Representations of the reals: Persistent homology is about studying sublevel sets of real-valued functions on topological spaces. Such sublevel sets have – for nice enough functions and spaces – discretizations that allow us to adapt descriptions of finite diagrams of vector spaces to efficient
descriptors. In particular, by using the “distance from a set” family of functions we can support inference of topological structures from point clouds.

So,

A dichotomy such as the one we have seen above cries out for a unifying theory…

Something that might be worth considering from this research is their graded R[x]R[x]-module structure on the chain complexes for different ϵi\epsilon_i, where RR is a PID. The action of xx is to move to the next ϵi\epsilon_i (p.6 of Barcodes).

Re: A Survey of Magnitude

Yes, there are all sorts of apparent resonances between magnitude and persistent homology. I mentioned a few of them in a talk last month at an applied topology meeting. The last slide says something about where I think some of the connections may lie.

One small thing worth noting is that the scale factors used in magnitude and persistent homology go in opposite directions. Let AA be a finite metric space. If we’re considering tAt A (as we do when we use the magnitude function t↦|tA|t \mapsto |t A|), then small values of tt mean that we’re shrinking AA or viewing it from far away, so that the points coalesce. But if we’re considering the Vietoris–Rips or Čech complex at scale ε\varepsilon, then it’s large values of ε\varepsilon that cause the points to coalesce.

For me it’s not so much a case of “catching up with” the persistent homology literature as getting the basics straight. There are some good introductions out there, some of which you’ve mentioned, and I’d like to work my way through one of them systematically.

Re: A Survey of Magnitude

I suppose we might also think of Laplace transforms.

On another note, do I recall somewhere Tom explaining the difference of the magnitude of a directed arc from that of an undirected on? Of course Mike’s homology theory works on directed simplices with asymmetric distances.

So, the magnitude of the nonsymmetric version is larger. In a way, that shouldn’t be surprising. If you pick a random pair of points on the line, then in the nonsymmetric version, there’s a 50 percent chance that the distance from the first to the second will be the ordinary Euclidean distance, but also a 50 percent chance that it will be ∞\infty. So, points are “typically further apart” than in the ordinary, symmetric version. And since magnitude is meant to be a measure of size, it makes sense that the magnitude is larger. This explanation even hints at where the factor of 2 comes from.

Re: A Survey of Magnitude

Anyway, is anyone else getting a sense that path integrals are in the air? After all, these chains in Mike’s homology are just paths.

Perhaps my bells have been primed to ring from the kinds of discussion at the Café from around 8 years ago. Urs Schreiber was certainly interested at the time in Tom’s measure, as you can see in this paper.

There in the same sentence are Tom and ‘path integral’ (perhaps not such a frequent occurrence):

Maybe the universal categorical properties of Kan extension in conjunction with a generalization of the Leinster measure can help us to finally nail the right definition of the (non-discrete) path integral?

Re: A Survey of Magnitude

What happens asymptotically when looking at the magnitude function this way around, where instead of imagining the space being scaled by tt, |tA||t A|, we think of the space as fixed and the weighting changing?

Now if AA is a compact subset of ℝn\mathbb{R}^n, we’re interested in the behaviour of Mt(A)M_t(A) as t→∞t \to \infty, where Mt(A)=sup{Mt(A′):A′⊆A,A′finite}M_t(A) = sup\{M_t(A^'): A^' \subseteq A, A^' finite\}.

Say, I’m looking to investigate point 5 of the post, so want limt→∞Mt(A)tnlim_{t \to \infty}\frac{M_t(A)}{t^n}. Can I swap the limits and consider sup{limt→∞Mt(A′)tn:A′⊆A,A′finite}sup\{lim_{t \to \infty}\frac{M_t(A^')}{t^n}: A^' \subseteq A, A^' finite\}?

Might it help to look further into the composition of χ(CℓA′)\chi(C^{A^'}_\ell) as a sum?

Re: A Survey of Magnitude

I guess another wrinkle to add would be what Mike’s discussing above of using intervals II to cover a range of ℓ\ell values. Presumably here we’d only care about IIs of the form [0,ϵ][0, \epsilon], for small ϵ\epsilon. But then perhaps we’d need to have a further limit as ϵ→0\epsilon \to 0.

Re: A Survey of Magnitude

Oh yes, of course it couldn’t be like that. It’s a balancing act between the increasing number of simplices with ever finer grids and the heavy hand of that exponential weighting.

Is there an intuitive way to reduce to the number of points in ever finer cubical grids of spacing 1/t1/t as the finite A′A^'s? That would produce tnt^n points in a unit cube, and if the exponential weighting could be taken to exclude the 1-simplices and higher, all would be fine.

Re: A Survey of Magnitude

Well, it would be fantastic if a method could be found that didn’t require serious amounts of analysis! And it would hardly be the first time that a geometric fact was proved initially by analysis, then later by homological methods.

Re: A Survey of Magnitude

From the series perspective, t→∞t\to\infty means q=e−t→0q=e^{-t}\to 0, so modulo issues of convergence this limit should be just looking at the constant term for ℓ=0\ell=0. The simplicial set MS(X,0)MS(X,0) is discrete on the points of XX, so its homology is concentrated in degree 0 with rank being the cardinality of XX. At least for XX finite this gives the right answer for limt→∞Mt(A)\lim_{t\to\infty} M_t(A). It’s less clear what to do with Mt(A)/tnM_t(A)/t^n from this perspective though.

Re: A Survey of Magnitude

Yes, I made this point back here and was trying to apply it way too naively just above.

I wonder if it might be possible to work things out with as pleasant a choice of those finite subsets as possible. I mean, imagine we use cubical meshes of ever finer spacing in a unit box in ℝn\mathbb{R}^n. Can we then get a grip on the number of low dimensional simplices?

Re: A Survey of Magnitude

Based on Richard’s point that natural chain homotopies are preserved by −⊗VA-\otimes_V A, here is a proposed proof that magnitude homology is invariant under equivalence of VV-categories. In fact, I will prove the more general statement that, as is the case for ordinary homology and Euler characteristic of ordinary categories, two VV-functors connected by a VV-natural transformation induce the same map on homology. Thus, magnitude homology is even invariant under adjunctions.

Let F,G:C⇉DF,G:C\rightrightarrows D be two VV-functors and α:F→G\alpha:F\to G a VV-natural transformation. It will suffice to show that the induced maps MC(C)⇉MC(D)MC(C) \rightrightarrows MC(D) are naturally chain homotopic (where MC(C),MC(D):Vop→ChMC(C),MC(D):V^{op}\to Ch). For this, it will suffice to show that the induced maps MS(C)⇉MS(D)MS(C) \rightrightarrows MS(D) are naturally simplicially homotopic (where MS(C),MS(D):Vop→sSetMS(C),MS(D):V^{op}\to sSet).

By definition, a simplicial homotopy between two simplicial maps f,g:X⇉Yf,g:X\rightrightarrows Y is a map X×Δ1→YX\times\Delta^1\to Y restricting to f,gf,g on the vertices. Since we know exactly what the nn-simplices of Δ1\Delta^1 are, this can be written out very explicitly in terms of functions from the sets XnX_n to the sets YnY_n; see for instance May’s Simplicial objects in algebraic topology. Using α\alpha we should be able to write down such a homotopy explicitly by postcomposing with maps between the objects C(xn−1,xn)⊗⋯⊗C(x0,x1)C(x_{n-1},x_n)\otimes \cdots \otimes C(x_0,x_1) and D(yn−1,yn)⊗⋯⊗D(y0,y1)D(y_{n-1},y_n)\otimes \cdots \otimes D(y_0,y_1) induced by the action of F,GF,G and the components αx:1→D(Fx,Gx)\alpha_x:1\to D(F x,G x). The result will then be natural by associativity of composition in VV.

Here’s an attempt at writing this out explicitly in somewhat more abstract language. Let ♯C\sharp C denote the codiscrete simplicial set on the objects of CC, with exactly one nn-simplex for every (n+1)(n+1)-tuple of objects of CC, and let el♯C\el\sharp C denote its category of elements. Then there is a functor C⊗:el♯C→VC^\otimes : \el\sharp C\to V sending (x0,…,xn)(x_0,\dots,x_n) to C(xn−1,xn)⊗⋯⊗C(x0,x1)C(x_{n-1},x_n)\otimes \cdots \otimes C(x_0,x_1) (and, in particular, (x0)(x_0) to the unit/terminal object). The functor MS(C)MS(C) can be obtained by composing this with the Yoneda embedding to get Y∘C⊗:el♯C→SetVopY \circ C^\otimes : \el\sharp C\to Set^{V^{op}}, then left Kan extending along the discrete opfibration πC:el♯C→Δop\pi_C:\el\sharp C \to \Delta^{op} to get a functor LanπC(Y∘C⊗):Δop→SetVopLan_{\pi_C} (Y\circ C^\otimes) : \Delta^{op} \to Set^{V^{op}}, which rearranges to a functor Vop→SetΔop=sSetV^{op}\to Set^{\Delta^{op}} = sSet.

An object of el(♯C×Δ1)\el(\sharp C \times \Delta^1) consists of an (n+1)(n+1)-tuple (x0,…,xn)(x_0,\dots,x_n) of objects of CC, together with an nn-simplex of Δ1\Delta^1. The latter is an order-preserving map θ:{0,…,n}→{0,1}\theta:\{0,\dots,n\} \to \{0,1\}, which amounts to a choice of some 0≤j≤n+10\le j \le n+1 such that θ−1(0)={0,…,j−1}\theta^{-1}(0) = \{0,\dots,j-1\} and θ−1(1)={j,…,n}\theta^{-1}(1) = \{j,\dots,n\}. Inspecting the definition of el♯α\el\sharp\alpha, we see that α⊗\alpha^\otimes at such an object is supposed to be a map from C(xn−1,xn)⊗⋯⊗C(x0,x1)C(x_{n-1},x_n)\otimes \cdots \otimes C(x_0,x_1) to

Finally, if we denote πC×Δ1:el(♯C×Δ1)→Δop\pi_{C\times\Delta^1} : \el(\sharp C \times \Delta^1) \to \Delta^{op}, then we have a simplicial natural transformation LanπC×Δ1(Y∘α⊗)Lan_{\pi_{C\times\Delta^1}}(Y\circ \alpha^\otimes). Its codomain is LanπC×Δ1(Y∘D⊗∘(el♯α))Lan_{\pi_{C\times\Delta^1}}(Y\circ D^\otimes \circ (\el \sharp \alpha)), which as before maps to πD(Y∘D⊗)\pi_D(Y\circ D^\otimes) by the mate of the equality πC×Δ1=πD∘el♯α\pi_{C\times\Delta^1} = \pi_D \circ \el \sharp \alpha. And its domain is LanπC×Δ1(Y∘C⊗∘elp)Lan_{\pi_{C\times\Delta^1}}(Y\circ C^\otimes \circ \el p), which since πC×Δ1=πC∘elp\pi_{C\times\Delta^1} = \pi_C \circ \el p can be identified with LanπCLanelp(Y∘C⊗∘elp)Lan_{\pi_C} Lan_{\el p}(Y\circ C^\otimes \circ \el p).

Now elp\el p is a discrete opfibration (since it is a map between two discrete opfibrations over Δop\Delta^{op}) whose fiber over an nn-simplex (x0,…,xn)∈♯C(x_0,\dots,x_n) \in \sharp C is just the set (Δ1)n(\Delta^1)_n of nn-simplices in Δ1\Delta^1. Thus, for any X:el♯C→EX:\el\sharp C \to E, the functor Lanelp(X∘elp)Lan_{\el p}(X \circ \el p) takes (x0,…,xn)(x_0,\dots,x_n) to the copower X(x0,…,xn)⋅(Δ1)nX(x_0,\dots,x_n) \cdot (\Delta^1)_n. And since coproducts commute with copowers, it follows that LanπCLanelp(Y∘C⊗∘elp)Lan_{\pi_C} Lan_{\el p}(Y\circ C^\otimes \circ \el p) is MS(C)×Δ1MS(C) \times \Delta^1. Therefore, α⊗\alpha^\otimes gives rise to a natural simplicial homotopy between MS(F)MS(F) and MS(G)MS(G).

(Lurking somewhere in here is a notion of “Segalic homotopy VV-enriched category” along the lines of Tom’s paper mentioned here using “colax functors” defined on el♯C\el\sharp C, with C⊗C^\otimes, F⊗F^\otimes, and α⊗\alpha^\otimes being how we regard ordinary VV-categories as homotopy ones. I feel like I’ve seen such a definition written down somewhere, but at the moment I can’t remember where.)

Re: A Survey of Magnitude

Excellent. Previously we knew that two VV-categories have the same magnitude if they’re connected by an adjunction. Now we know that they have the same homology if they’re connected by an adjunction. So, you’ve categorified the existing result.

Better still, the magnitude result needs the hypothesis that both categories have well-defined magnitude, but that hypothesis disappears when you move to the categorified level.

Based on Richard’s point that natural chain homotopies are preserved by −⊗VA- \otimes_V A

I didn’t follow all the twists and turns of your conversation with Richard, but do I understand correctly that for any functor A:V→ChA \colon V \to Ch (not just A:V→AbA \colon V \to Ab), the functor

−⊗VA:[Vop,Ch]→Ch
- \otimes_V A \colon [V^{op}, Ch] \to Ch

sends natural chain homotopies to chain homotopies? If I’m understanding, this is a simple calculation using only:

the fact that a chain homotopy between chain maps A⇉BA \rightrightarrows B amounts to a map I⊗A→BI \otimes A \to B (where II is a certain chain complex);

the fact that tensor product of chain complexes distributes over coends;

the associativity of ⊗\otimes.

It follows that if two functors F,G:Vop→ChF, G \colon V^{op} \to Ch are naturally chain homotopy equivalent, then F⊗VAF \otimes_V A and G⊗VAG \otimes_V A are chain homotopy equivalent. In particular, this applies when FF and GG are the unnormalized and normalized versions of MCMC, resolving the earlier question. Right?

I’m not attempting to say anything new here; I just want to verify that I’ve followed everything correctly.

Re: A Survey of Magnitude

Yes, that is an accurate overview of the argument that I gave in that part of the discussion.

In a parallel discussion, I am attempting to find a point of view on Mike’s construction of a homology theory that fits into a general theory of cohomology of enriched categories, but there is some way to go here :-).

Re: A Survey of Magnitude

…that you asked elsewhere in this thread. The point that I find unsatisfactory about your construction is the following: if I have a category enriched over some VV, how do I know what V′V' should be?

In the case that VV is SetSet, we take V′V' to be 11. But sometimes we take VV just to be VV. What is the general recipe? What should V′V' be for TopTop, AbAb, chain complexes, simplicial sets, CatCat, etc?

What I would hope for is a recipe which, given any VV-enriched category (under some reasonably general assumptions on VV), provides with me a cohomology/homology theory, just like for ordinary categories. Unless I’m missing something, your construction does not provide this, for the above reason. Even for small VV, I’m not convinced that we have enough evidence to conclude that it is ‘correct’ to always take V′V' to be VV.

This is not to put down your construction at all, it is just to explain why I am looking for a different point of view on it.

It is plausible to me that there might be a quite general theory of homology/cohomology for enriched categories. I will try to outline a few more thoughts on this elsewhere in the discussion.

For the kind of VV used for magnitude homology, we might replace VV itself with the presheaf category upon it (with its Day convolution monoidal structure). This is sort of the opposite direction from your construction, in which you replace the category of presheaves on 11 by 11.

Re: A Survey of Magnitude

Mike wrote:

I consider my construction to be a general theory of (co)homology of enriched categories… (-:

I agree!

As far as I’m concerned, the only fly in the ointment is the semicartesian requirement (or alternatively, the requirement of an augmentation). It’s unclear to me whether that requirement has to be there for some principled reason, or whether we could bypass it by shifting viewpoint somehow.

In any case, there are lots of semicartesian/augmented monoidal categories that are interesting to enrich over, so we do have a pretty general (co)homology theory of enriched categories.

Richard wrote:

The point that I find unsatisfactory about your construction is the following: if I have a category enriched over some VV, how do I know what V′V' should be?

I believe this is a misunderstanding of the part of Mike’s comment where V′V' was introduced. Let me have a go at explaining.

Here’s the set-up. We have a monoidal category VV (maybe not small), a small VV-category CC, and a functor A:V→AbA\colon V \to Ab (or V→ChV \to Ch). From CC we’ve constructed a functor MC:Vop→ChMC\colon V^{op} \to Ch. We now want to construct the chain complex

It certainly does if VV is small, since ChCh has small colimits. More generally, it also exists if VV and AA have the property that there is some small full subcategory V′V' of VV such that AA is the left Kan extension of A|V′A|_{V'} along the inclusion V′↪VV' \hookrightarrow V. Under that hypothesis, MC⊗VAMC \otimes_V A exists and is equal to MC|V′op⊗V′A|V′MC|_{V'^{op}} \otimes_{V'} A|_{V'} for any such V′V'.

(Mike used slightly different notation, writing A′A' for the restriction A|V′A|_{V'}. The key point is that a Kan extension along a full and faithful functor is always a genuine extension, i.e. makes the usual triangle commute. So once you’ve chosen your subcategory i:V′↪Vi \colon V' \hookrightarrow V, there’s only one possible functor A′:V′→ChA' \colon V' \to Ch such that A=LaniA′A = Lan_i A', namely, A|V′A|_{V'}.)

So, the chain complex MC⊗VAMC \otimes_V A and its homology H*(C;A)H_\ast(C; A) don’t depend on a choice of V′V'; they only depend on the existence of V′V'. If at least one such V′V' exists, it doesn’t matter which one you use. They all produce the same homology.

This condition on AA is very similar to the notion of “small presheaf”. For a category ℰ\mathcal{E}, a presheaf A:ℰop→SetA \colon \mathcal{E}^{op} \to Set is said to be small if it can be expressed as a small colimit of representables, or equivalently if there is some small full subcategory ℰ′\mathcal{E}' of ℰ\mathcal{E} such that AA is the left Kan extension of A|ℰ′opA|_{\mathcal{E}'^{op}} along the inclusion ℰ′op↪ℰop\mathcal{E}'^{op} \hookrightarrow \mathcal{E}^{op}. If ℰ\mathcal{E} is small then all presheaves on ℰ\mathcal{E} are small, but that’s false when ℰ\mathcal{E} is large. Generally, the category of small presheaves on ℰ\mathcal{E} is the free cocompletion ℰ^\widehat{\mathcal{E}} of ℰ\mathcal{E}. This defines a monad ^\widehat{\,\,\,} on the category of locally small categories.

Re: A Survey of Magnitude

I believe this is a misunderstanding

Thanks for the elaboration! But it was not in fact a misunderstanding, at least not in the way that you discuss (the use of these ideas in the free co-completion of a large category was indeed the first thing that came to mind when I read Mike’s post) :-). What I find unsatisfactory is the use of V′V' at all. One can define the nerve of a category perfectly well without these kind of tricks; why not the nerve of an enriched category? If I am provided with a VV, I do not typically think that I will be able to cook up in a nice, categorical way (i.e. just from knowing some of its categorical properties) the appropriate V′V' for a given AA, or a fixed V′V' which determines some collection of AA’s.

As I wrote, but maybe did not put clearly, I am not even sure that one of the premises of your elaboration is correct: that when VV is small, then we do not need to do anything. To emphasise, this ‘correctness’ is not an objection to being able to carry out the construction; it concerns whether or not we are likely to obtain a useful cohomology theory. How do we know that it might not be better to restrict to some smaller V′V' for a given small VV, in some situation? Hence my question about (the lack of) a general recipe.

I do, in addition, think that semi-cartesianness should not be needed (one should certainly be able to say something about enrichment other categories such as AbAb); and I also, from an aesthetic point of view, am not entirely keen on the overall approach of cooking up a functor Vop→sSetV^{op} \rightarrow sSet from a given VV-category, and feel that there might be better ways to ‘see’ the VV-enrichedness. But the point about the use of V′V' seems to me to be the most significant of these objections.

Of course, though, to have any construction at all which produces some interesting examples is great! This discussion is only about the broader question you asked about understanding cohomology of enriched categories.

Re: A Survey of Magnitude

I’m afraid I just don’t know what you mean. You previously wrote things like:

if I have a category enriched over some VV, how do I know what V′V' should be?

and

What is the general recipe? What should V′V' be for TopTop, AbAb, chain complexes, simplicial sets, CatCat, etc?

I replied by saying that these questions don’t make sense. (Sorry, that sounds rude, but it’s not meant to be — I’m just trying to be clear.) For a given VV, there is no single thing called “V′V'”. The magnitude homology H*(C;A)H_\ast(C; A) of a VV-category CC with coefficients in A:V→ChA\colon V \to Ch depends on VV, CC and AA only. There is no “V′V'” that it also secretly depends on. There is no part of the construction in which you have to choose a V′V'.

So I’m puzzled that you say you hadn’t misunderstood that point, because that seems incompatible with what you wrote previously and some of the things you just wrote.

Let me have another go. A functor A:V→ChA\colon V \to Ch may be said to be small if it is the left Kan extension of its restriction to some small full subcategory. Mike defined a graded abelian group H*(C;A)H_\ast(C; A) for each monoidal category VV, small VV-category CC, and small functor A:V→ChA \colon V \to Ch.

And that’s that. The symbol V′V' didn’t even appear in the previous paragraph.

the appropriate V′V' for a given AA

I’d quibble with the word “the”. The functor AA is either small or not. If it is, then by definition it’s equal to the left Kan extension of A|V′A|_{V'} for some small subcategory V′⊆VV' \subseteq V. But in that case, the same is true when V′V' is replaced by any larger small subcategory of VV. So if AA is small, there are in general lots of small subcategories V′V' with the property just stated.

It’s true that given a functor A:V→ChA \colon V \to Ch, it might be hard to determine whether or not AA is small, and, if it is, to find a small V′V' such that AA is the left Kan extension of A|V′A|_{V'}. But that’s nothing especially to do with magnitude homology. It’s not even especially to do with functors or category theory. Similar issues arise all over algebra: e.g. given a group AA, it might be hard to determine whether or not AA is finitely generated, and, if it is, to find a finite Γ\Gamma such that AA is generated by Γ\Gamma.

it concerns whether or not we are likely to obtain a useful cohomology theory. How do we know that it might not be better to restrict to some smaller V′V' for a given small VV, in some situation?

Again, I don’t know what you mean. The (co)homology is the same no matter what V′V' you might have used in order to prove that the functor of coefficients AA is small.

What I find unsatisfactory is the use of V′V' at all. One can define the nerve of a category perfectly well without these kind of tricks; why not the nerve of an enriched category?

To answer your question more broadly, why might we expect it to be non-trivial to generalize the notion of nerve from ordinary categories to enriched categories? Because for ordinary categories (small ones, at least), the totality of objects and the totality of arrows are the same type of thing: sets. But for enriched categories, they’re not: the totality of objects is a set, whereas the totality of arrows is… well, one person might say it’s a set-indexed family of objects of VV, and another (taking the coproduct of that family) might say it’s a single object of VV, but whatever it is, it’s not a set. And the fact that the totality of objects has a different type from the totality of arrows is bound to make matters harder.

Re: A Survey of Magnitude

For the sake of other readers, present or future, I would like to clarify once more what I am saying. It may not be something that you or Mike feel is significant, but it is a perfectly valid point about the construction, it is one that I feel is significant, and it is not one based on misunderstanding.

You wrote the following.

It’s true that given a functor A:V→Ch(A)A:V&#8594;Ch(A), it might be hard to
determine whether or not AA is small, and, if it is, to find a small V′V' such that
AA is the left Kan extension …

That is exactly the point that I am making. I do not find this satisfactory. Where I wrote ‘what should V′V' be’, and ‘what is the general recipe’, I was referring exactly to two questions that come immediately out of what you wrote: what is a general recipe for determining whether AA is small, and what is a V′V' that comes out of that recipe for a given AA? You are agreeing that these questions are valid; in particular, they make perfect sense!

I agree completely that there is not a unique V′V' in a given setting. Nothing that I wrote was intended to suggest that there was.

I’d quibble with the word “the”.

What I meant, as above, was: what is the V′V' that comes out of some specific general recipe for constructing such a category given VV and AA? Again, I am not at all suggesting that there is only one possible V′V', nor that, if there were any general recipe, that it would be unique.

The magnitude homology depends on VV, CC and AA only. There is
no “V′” that it also secretly depends on. There is no part of the construction in
which you have to choose a V′V&#8242;.

I understand that completely. As I have said already several times, it is the treatment of the case that VV is SetSet that is more relevant for the point I am making. Here we do have to choose a V′V', and this is by no means an isolated case. As again I have already said, in this discussion I am not really thinking about magnitude homology at all, except as one of several examples; I am thinking about cohomology of enriched categories in general. And in general, if one considers Mike’s construction to be a homology/cohomology theory of enriched categories, the existence of a V′V' is part of the construction (and thus, as category theorists, we would probably instead postulate it as part of the data, if one is formulating things in a purely general context).

Again, I don’t know what you mean. The (co)homology is the same no matter
what V′V&#8242; you might have used in order to prove that the functor of coefficients
AA is small.

My point is that one can turn things around. Rather than begin with AA, one can fix a choice of V′V', and use only those coefficients which arise as left Kan extensions using this choice. Mike’s original post had in fact, to my reading at least, more of this flavour than the flavour you are emphasising (in which one fixes AA and then tries to find a relevant V′V'). If V′V' is part of the data in general (which, as I wrote above, is a reasonable thing to do when formulating Mike’s construction generally), it is part of the data when VV is small too, and one can perfectly well choose it to be something other than VV itself.

Because for ordinary categories (small ones, at least), the totality of objects and
the totality of arrows are the same type of thing: sets.

I’ve already addressed this point too. One has to take into account the enrichedness somehow, that is clear. That we have to rely on smallness tricks does not follow at all, it is only something that happens to arise in Mike’s construction. It is necessary in Mike’s construction to use the trick even in the case of enrichment over SetSet, whereas no such trick is necessary a priori, as we all know.

Re: A Survey of Magnitude

it is the treatment of the case that VV is SetSet that is more relevant for the point I am making. Here we do have to choose a V′V'

No, I don’t see why we would have to choose a V′V' in that case any more than in any other case. It’s true that by allowing arbitrary V′V' we allow more general coefficients than “classical” homology of categories does, but I don’t see anything wrong with that.

and thus, as category theorists, we would probably instead postulate it as part of the data, if one is formulating things in a purely general context

No, I don’t agree. Why would we do that? Category theorists can and do use (−1)(-1)-truncation just like anyone else.

Rather than begin with AA, one can fix a choice of V′V', and use only those coefficients which arise as left Kan extensions using this choice.

Re: A Survey of Magnitude

I don’t wish to discuss this point endlessly. I think you tacitly are acknowledging here that you do see what I am getting at, even if you do not agree that it is significant. Hopefully the same is true of Tom.

No, I don’t see why we would have to choose a V′V&#8242; in that case any more than in any other case.

Surely it is clear what I had in mind here? What I meant is that, given an AA, one has to, in the case of SetSet, make use of (use whatever verb you like best here) some V′V' that is not VV itself. And that for me is unsatisfactory; it is not necessary in order to construct the resulting homology theory, there is no canonical way to obtain some V′V' that works, etc.

No, I don’t agree. Why would we do that? Category theorists can and do use
(-1)-truncation just like anyone else.

There are two ways one could state your construction. The first goes: suppose that we have a functor A:V→ChA :V \rightarrow Ch, and that there exists a full subcategory V′V' of VV such that AA is the left Kan extension along the inclusion of V′V' in VV of its restriction to V′V'. Then, picking some such V′V', we …

The second goes: suppose that we have a functor A:V→ChA: V \rightarrow Ch, and a full subcategory V′V' of VV such that AA is the left Kan extension along the inclusion of V′V' in VV of its restriction to V′V'. Then we …

I prefer the second formulation, and I suspect many category theorists would. For a start, the logic needed to express it is much weaker.

Of course, one is perfectly at liberty to choose the first formulation, and if one is determined to be obstinate, no doubt we could enter in a lengthy discussion about it. I’ll not go in for that, though :-).

I’ll not respond specifically to the other two comments, which I think were unnecessary. The fact that I have had to explain at length what to me is a simple observation, to convince you and Tom that I am not just misunderstanding something, should be evidence enough that the flippancy is not really warranted; one can accept that somebody might find an aspect of a construction unsatisfactory on reasoned grounds, and that somebody might not consider that construction to be the final word on the subject, without agreeing with those things oneself.

Re: A Survey of Magnitude

I’m sorry, that comment was supposed to be humorous, but I’m not good at making jokes so it came out as rude. My apologies. But I really don’t know what else to say. You say you don’t like having to choose a V′V'. Fine; we described a version of the construction where you don’t have to choose a V′V'. Now you seem to be saying that you prefer to choose a V′V'. I really don’t know what to say other than “make up your mind!” (-:

Anyway, this conversation has probably reached the point of diminishing (or zero) returns, but in case it helps, let me point out that one could also take the point of view that the tensor product, and hence the homology, always exists, but perhaps in a larger universe. Some coefficients happen to have the property that the corresponding homology is always small, whatever the category. But some categories also have the property that their homology is small, whatever the coefficients: for instance, unless I’m mistaken the unit VV-category II satisfies H*(I;A)=A(1)H_\ast(I;A) = A(1) (in degree 0) for all A:V→AbA:V\to Ab. So from this perspective “smallness” is just a property of the homology that one can ask about and investigate, analogous to finiteness or finite-generation for ordinary homology.

Re: A Survey of Magnitude

It’s moot now no doubt with the latest developments, but my point of view was fundamentally simply that I was objecting to needing to use some V′V' at all. To try to give some feeling for this objection, I was also trying to intimate that this use of V′V' might be of a little deeper significance than it might first appear, in that a priori we could fix one choice, and this would define which coefficient systems we allow; which seems strange to me, as maybe only those coefficient systems for certain choices of V′V' would give interesting homology theories. I do now see how this juxtaposition could be bewildering :-)!

Re: A Survey of Magnitude

Richard, let me say out loud — in case of doubt — that I very much appreciate your contributions to this thread. You’ve made substantive mathematical points, and (at a more basic level) it also makes a huge difference to have people actually participating. It looks like there are only three people on the planet who’ve been following this thread in detail, and you’re one of them.

I say this because obviously this conversation has got slightly fractious, as often happens on web forums when there’s miscommunication. Once there have been a couple of rounds of misunderstanding and frustration, it’s natural for participants to state things in more and more stark terms in an effort to be crystal clear. I was aware that I was doing that back here, which is why I apologized for any appearance of rudeness and said that I was simply aiming for clarity. Still, this sort of conversation is frustrating for all concerned, even when everyone is confident that everyone else is acting in good faith, as I believe we all are here.

As for the actual issues, I now understand better what you were saying, and for the moment I have nothing more to say on that front.

Re: A Survey of Magnitude

Thanks very much to you both for the generous remarks, here and on the new thread! I of course indeed understood that everybody was acting in good faith. Frustration can occur face to face as well when trying to convey something!

I very much agree that simply participating in a discussion like this is enormously under-valued.

As noted, focusing on the smallness seems now to have led to some interesting progress, which is great!

Re: A Survey of Magnitude

I have very little to add here; Tom is making pretty much exactly the points that I would. However, I did want to say again that I’m much less bothered by the semicartesian/augmentation hypothesis now that I see the relevant “nerve” as a two-sided bar construction.

In general, for any VV, any VV-category CC, and any VV-functors F:Cop→VF:C^{op}\to V and G:C→VG:C\to V, there is a 2-sided simplicial bar construction B•(F,C,G)∈VΔopB_\bullet(F,C,G) \in V^{\Delta^{op}} defined by

This is a very well-known and well-studied thing in homotopy theory. If V=SetV=Set, then taking FF and GG constant at the 1-point set, the simplicial set B•(1,C,1)∈SetΔopB_\bullet(1,C,1) \in Set^{\Delta^{op}} is just the ordinary nerve of CC. Thus, in general B•(F,C,G)B_\bullet(F,C,G) can be regarded as a sort of “nerve of CC relative to FF and GG”. It’s also the “homotopy tensor product” of FF and GG over CC, and the homotopy colimit of GG weighted by FF. (In particular, the nerve of CC is the conical homotopy colimit of the CC-diagram constant at 1.)

When VV is semicartesian, the “parametrized nerve” of a VV-category that we’re using for magnitude homology is obtained simply as follows:

Regard a VV-category CC as a SetVopSet^{V^{op}}-category C^\hat{C} by the Yoneda embedding of VV, which is strong monoidal by Day convolution.

Take FF and GG constant at the terminal/unit object and form the 2-sided simplicial bar construction B•(1,C^,1)B_\bullet(1,\hat{C},1).

A bit more generally, if VV is not semicartesian but CC is augmented, then the augmentation can be used to define functors FF and GG that are constant at the unit object, yielding a similar B•(I,C^,I)B_\bullet(I,\hat{C},I). But even without semicartesianness or augmentation, there are things we can do. For instance, we can always take FF and GG to be constant at the terminal object, even if it doesn’t coincide with the tensor unit. (If there is a zero object like in AbAb, this version is boring since 0⊗A≅00\otimes A \cong 0 for any AA; but in other cases it might be interesting.) For a particular VV, or even a particular CC, there may be other useful choices of FF and GG.

Re: A Survey of Magnitude

In this generality as homology from a two-sided bar construction, what was already known?

I see at nLab: simplicial bar construction, there’s a similar simplicial complex occurring, except the end points seem to be fixed. (As I said above, it all feels rather path-integral-ish.) Did no-one think then to derive a homology?

Re: A Survey of Magnitude

The fixed-endpoint version is just the case where FF and GG are representables. And of course people take homology of simplicial bar constructions all the time. So I think the main new thing here is applying the Yoneda embedding of VV first, allowing us to then use coefficients of the form A:V→AbA:V\to Ab.

Re: A Survey of Magnitude

Could I ask what the status is of the inclusion-exclusion formula |X∪Y|=|X|+|Y|−|X∩Y||X \cup Y| = |X|+|Y|-|X \cap Y|? I thought this was the most surprising property of magnitude, especially since it is true so often but not always.

I see Conjecture 4.5, (false) Equation 4.13 and open question (5), but it isn’t clear to me what you now think the larger picture is. When should I expect this formula to hold?

Re: A Survey of Magnitude

I’m fighting a catastrophically losing battle to stop thinking about magnitude and do the other things I really have to get done before semester starts, but apparently I can’t resist writing one more reply.

I think your question is a very good one. When we believed the convex magnitude conjecture, we of course believed that the inclusion-exclusion principle

Now we know that the convex magnitude conjecture is false. In fact, we know that the inclusion-exclusion principle fails for convex sets. For if it held, then magnitude would be a monotone, isometry-invariant valuation on convex sets, which (basically by Hadwiger’s theorem) would imply that it was a linear combination of the intrinsic volumes (as conjectured). That in turn would imply that for any convex XX, the magnitude function t↦|tX|t \mapsto |t X| is a polynomial. And we know that when XX is a ball of odd dimension ≥5\geq 5, it’s not.

I can think of two situations where inclusion-exclusion definitely does hold. They both have a “1-norm” flavour about them.

Let AA and BB be subsets of a metric space XX. Say that AAprojects to BB if for all a∈Aa \in A, there exists π(a)∈A∩B\pi(a) \in A \cap B such that
d(a,b)=d(a,π(a))+d(π(a),b)
d(a, b) = d(a, \pi(a)) + d(\pi(a), b)
for all b∈Bb \in B. If XX is positive definite, and AA and BB are compact with either AA projecting to BB or vice versa, then the inclusion-exclusion formula holds for AA and BB.

This is a pretty restrictive condition which pretty much never holds in Euclidean space. It holds much more often in ℓ1n\ell_1^n, which is why I say it has a 1-norm flavour about it.

The convex magnitude conjecture is still open in ℝn\mathbb{R}^n for n≤4n \leq 4. There’s some evidence that it’s true in ℝ2\mathbb{R}^2, namely, Simon’s numerics. These strongly suggest that the conjecture holds for various regular shapes such as squares and disks (and also the 3-cube). The numerical experiments are much more computationally feasible in low dimensions.

It would be good to test the conjecture for less regular shapes. E.g. you could do the following:

Choose at random 10 points in ℝ2\mathbb{R}^2.

Let XX be their convex hull, a convex polygon.

Choose a fine mesh MM of points approximating XX.

Compute |tM||t M| for several values of tt, thus obtaining an approximation to the magnitude function of XX.

See whether this approximation is close to the quadratic predicted by the convex magnitude conjecture.

As far as I know, no one’s done this.

I thought [inclusion-exclusion] was the most surprising property of magnitude, especially since it is true so often but not always.

I remember us discussing this kind of thing years ago, and I remember you had that example of a bent line. But I’ve forgotten why you found inclusion-exclusion so surprising. Can you remind me?

Re: A Survey of Magnitude

Tom, I took up your challenge. It was fun thinking about convex hull algorithms and implementing them in Python; it was also good practice for when I start teaching Python in a couple of weeks time. I’ll put the Python code up if there is interest from anyone.

The results are very far from conclusive, and I would say they are murky. The conjecture states that the magnitude of a convex body looks like

area2π+semiperimeter2+1.
\frac{area}{2\pi}+\frac{semiperimeter}{2}+1.

There are heuristic arguments for this as well as numerical evidence. One nice thing about this form is that it means that as we shrunk the width of a rectangle to nothing, its magnitude converges on the magnitude of the interval of the appropriate length: take a rectangle of dimensions a×ba\times b, the conjectured value of the magnitude is ab/2π+(a+b)/2+1ab/2\pi + (a+b)/2 + 1, as b→0b\to 0 this conjectured magnitude converges on a/2+1a/2 + 1 which we know is the magnitude of the interval of length aa.

Now running my code, when the convex set is small, so that I get a decent approximation from my grid, I find that a reasonable fit to the data is

There are so many unknowns in the numerics and approximations that it’s difficult to know how seriously to take the 2.12.1 which is itself very rough. With justification from other heuristics as mentioned above, it’s easy to dismiss.

Anyway, it’s not going to be 2.12.1, is it? I mean what’s 2.12.1 other than 22 up to experimental error? Well thinking about nearby candidates for respectable mathematical constants I realised 2π/3≃2.0942\pi/3 \simeq 2.094.

This means that the numerical evaluations of the magnitude of random convex polygons in the plane appear to be reasonably approximated by

[Note: (1) I assume “open” means the closure of an open set, and (2) this formula seems to work for n=1n=1 as well.]

Now we are in the case of even nn, as n=2n=2, and non-smooth boundary, which their work does not cover, but assuming that their formula did hold in our case, then we are lead to the following behaviour as R→∞R\to \infty:

then as a rectangle narrowed to a length aa line as above we would get the magnitude tending to a/(2π/3)+1<a/2+1a/(2\pi/3)+1 \lt a/2 + 1 which contradicts, I believe, the lower semiconinuity that we know magnitude has.

So, in conclusion, it’s somewhat inconclusive. However, I wouldn’t be surprised now if the conjecture were not true. I look forward to seeing what can be figured out for even-dimensional convex sets.

Re: A Survey of Magnitude

That’s all very interesting — thanks.

I like to imagine that you got 2π/32\pi/3 from some published list of mathematically simple real numbers. As in, you’re getting roughly 2.1 from some numerical experiment, you type 2.1 into the website of The List, and it tells you that what you’ve really got is probably 2π/32\pi/3. The List would be like Ramanujan recognizing the significance of 1729, but automated.

Nevertheless, as you point out, that constant can’t be right, because 2π/3>22\pi/3 \gt 2 and magnitude is lower semicontinuous. As you change a space gradually, the magnitude can suddenly drop, as in your example of a finite metric space XX with the property that
limt→0|tX|→6/5>1=|0X|.
\lim_{t \to 0} |t X| \to 6/5 \gt 1 = |0 X|.
However, the magnitude can’t suddenly rise, and that’s what it would do if 2π/32\pi/3were right.

In your experiments, you’re seeing magnitudes that are slightly smaller than what we predicted. Of course (as you know very well, but I’m thinking out loud) that could just be because you haven’t used enough points when approximating the space. Are you using similar numbers of points to what you used for the disk and square? Have you tried it for a long thin rectangle (i.e. something similar to a line segment)?

Re: A Survey of Magnitude

Re: A Survey of Magnitude

That’s almost exactly what I was imagining!

I wonder if it’s possible to adjust the “resolution”? E.g. if you put in 5.859874482 you get the suggestion of π+e\pi + e (which really is that, to 9 decimal places), but if you put in 5.86 you get much more complicated suggestions.